Although the visitors gave themselves hope through Marco Reus's goal just three minutes into the second half, Valere Germain put the outcome beyond all doubt with a third for Leonardo Jardim's side in the 81st minute.
The result confirmed a 6-3 aggregate win for the Ligue 1 leaders and put them in the draw for the semi-finals alongside Real Madrid, Atletico Madrid and Juventus.
Here is how Sports Mole covered the action in the Principality.
6.48pmThomas Tuchel's charges travel to the principality looking to overcome a one-goal deficit against Monaco when they visit the Stade Louis II for this tie's decider.
6.50pmThe Ligue 1 side hold a 3-2 lead from Wednesday's rescheduled first leg at the Signal Iduna Park, Kylian Mbappe impressing with a brace. But the encounter was overshadowed by what had happened on the day before, when the game was originally supposed to be played.
6.52pmAs Dortmund were making their way to the stadium, three bombs detonated near the team bus, shattering windows, injuring defender Marc Bartra - who had to undergo arm surgery - and causing shock to the players.
6.54pmBut while the attack itself is being investigated by German police (and there continues to be debate between Dortmund and UEFA about being "forced" to play just 24 hours after the traumatic event), the focus now turns to the football, and who will join the two Madrid giants and one of Juventus and Barcelona in the semi-finals.
6.56pmWithout further ado, here are the team news from the Stade Louis II.
7.03pmLeonardo Jardim makes three changes to the Monaco starting XI from their 2-1 win over Dijon on Saturday as Joao Moutinho, Thomas Lemar and Radamel Falcao are drafted in from the off, while Fabinho misses out after being booked in the first leg and Nabil Dirar and Valere Germain make the bench.
7.06pmLikewise, Thomas Tuchel makes a a trio of tweaks from the weekend's 3-1 win over Eintracht Frankfurt. Marcel Schmelzer, Christian Pulisic and Sven Bender all drop to the bench, while Erik Durm, Matthias Ginter and Raphael Guerreiro make the starting XI in their place.
7.09pmDortmund have a mighty task ahead of them tonight - Monaco's tally of 23 home wins is the best in Europe's top five leagues, ahead of Juventus (22), Barcelona, Roma and Spurs (all 20).
7.11pmThis just in - Dortmund have tweeted that kickoff will be delayed by five minutes until 7.50pm. The club say the police held the team bus at the hotel for 20 minutes 'without any reason'.
7.15pmNot the ideal start to the night for Dortmund, whose efforts to get the right focus for this game must have been difficult enough as it is - we'll keep you up to date with any other developments, but as it stands, kickoff has been put back by five minutes.
7.17pmIn Tuesday's Dortmund press conference, Tuchel told reporters: "We're more stable emotionally. All that happened last week has made us stronger. Now we have to play well and I'm convinced we can do that. We're ready and focused. We know it will be very tough, but we have the energy and confidence we need to get a result."
7.20pmHere is UEFA's official confirmation of kickoff being put back.
Due to late team arrival, the quarter-final second leg between Monaco and Dortmund has been delayed by five minutes.#UCL
7.22pmSo Dortmund mentioned a police delay 'without any justification', but nothing of the like from UEFA, who only mention a "late team arrival".
7.24pmKylian Mbappe, who will be in action tonight from the start, has been nothing short of sensational for Monaco this season. Despite being just 18 years old, he already has 21 goals to his name this season in all competitions, as well as 11 assists, all from just 36 games. Four of these goals have come in seven Champions League games, including last week's brace at BVB. It would be a huge surprise if there weren't any scouts tonight keeping tabs on the young forward.
7.27pmWhoever makes it through the tie tonight will be joining Real Madrid, Atletico Madrid, and one of Juventus or Barcelona in the semi-finals. The draw for the semi-finals will be held on Friday, April 21 at 11am BST, with the first legs being played on May 2 and 3, and the second legs on May 9 and 10.
7.30pmAmid all the headlines surrounding this tie and what looks to be a tall order for Dortmund to make it past Monaco, there are two milestones that have been hit tonight by the German giants.
7.37pmMATCH STATS: All five previous teams winning 3-2 away from home in the first leg of their Champions League knockout tie have progressed to the next round.
7.39pmMonaco have won four of their last five home games against German opposition, drawing the other, in all competitions, and are in with a shout of making the final four of the CL for the fourth time after 1993-94, 1997-98 and 2003-04 - no other French team has made the semis than once.
7.41pmHowever, Monaco have conceded 15 goals in the Champions League this season, more than any other team among the remaining quarter-finalists.
7.42pmAs for Dortmund, they have lost eight of their last 11 games in the Champions League knockout stages, winning the remaining three - will they be making that record better or worse tonight?
7.44pmAlthough Marc Bartra is not playing tonight, due to recovering from his wrist surgery, he is in attendance tonight, and will also be addressing his Dortmund teammates in the final 30 seconds of the pre-match team talk.
7.45pmPREDICTION: Five minutes until kickoff and it's time for me to predict the score. I reckon Dortmund will be fully focused on the game and will have extra impetus to go out and get a result tonight, in the face of last week's events. I predict a 2-0 win for the visitors and progress to the semi-finals.
7.47pmBoth sets of players emerge out of the tunnel and onto the pitch of the Stade Louis II stadium - the Champions League theme rings out and there's a huge round of applause from both sets of fans as it ends.
7.48pmYou'll Never Walk Alone has been ringing out ahead of kickoff from both sets of fans - nice touch from Monaco.
4 minThe wonderkid does it again! His third goal in this tie comes just three minutes into the second leg as he finds himself perfectly placed to coolly slot home after Burki parried a long-range Mendy effort into the path of the 18-year-old!
5 minA big setback for Dortmund, who now need three goals on the night to progress.
8 minDortmund continue to look shakey at the back as Mendy's cross from the left almost reaches Falcao, but Ginter just about manages to clear.
9 minNow for a goalkeeping error at the other end as Subasic tries to take the ball around Aubameyang and gets tackled, but the stopper just about manages to shield the ball and clear.
10 minHere come Dortmund again as Durm, on the right flank, slips a ball inside for Reus, who lets rip towards goal from about 12 yards out, but it's straight at Subasic, who gathers.
13 minCHANCES! It's exhilarating, end to end stuff here - Toure whips a cross from the right which Falcao heads just wide, and not long after, Mendy lofts the ball towards Silva at the back post but the latter heads straight at Burki.
14 minOFF THE POST! The woodwork is rattled as Sahin's free kick curls over the Monaco wall and has Subasic beaten, but it ricochets off the post and bounces away before it's hacked clear.
18 minIs that the goal that puts Dortmund out of the Champions League? A textbook header from the rejuvenated Falcao, who rises highest to nod Lemar's cross home from 12 yards out. Now Dortmund need FOUR goals to progress.
19 minMbappe and Falcao have a frankly ridiculous 49 goals between them this season.
20 minDortmund are all over the place at the back. Ginter gave Falcao far too much space, allowing Lemar to pick him out with ease. Did the 20-minute delay in getting to the Stade Louis II unsettle the players just a bit too much tonight (on top of last week's events)?
22 minMonaco are nearly through again as Falcao and Mbappe link up, but the latter slips and the potential chance goes begging. They're in total control.
24 minDortmund's sloppy defending is nearly shown up once again as Mbappe forces a corner, which Lemar swings in and Durm nearly slices into his own net. He gives away another corner but the second one comes to nothing.
26 minSUBSTITUTION: An early chance from Tuchel as Erik Durm comes off and Dortmund's own young prodigy, 19-year-old Ousmane Dembele, comes on.
28 minOnce again Monaco go through the Dortmund defence with ease, Lemar getting into space and having all the time in the world to whip in a decent cross, but it's just too heavy for Falcao.
30 minDembele picks up the ball 30 yards out on the left and lofts a cross down the middle which nearly puts Aubameyang through, but Jemerson is there to head clear.
32 minDembele has certainly added a bit of spark to Dortmund's attack since coming on - he wins a free kick in a decent position, but Reus's effort is underhit and Monaco clear.
33 minMonaco maraud forward on the counter, three on two, and Silva, who could put Mbappe clear through down the right, instead hits the ball at Guerreiro, and Dortmund get it away.
34 minSahin snuffs out a counter-attack from Monaco, bears forward and looks for Aubameyang in the box, but his lofted ball is caught by Subasic.
37 minDortmund break again and Kagawa slips the ball through for Aubameyang, but the striker mistimes his run and the flag goes up for offside.
40 minCHANCES! Much better from the visitors, who are enjoying a bright spell now - first Dembele plays an incisive forward pass to Reus in the box, but he fails to find Aubameyang with a backheel which Subasic claims, and moments later, Guerreiro gets the ball inside the box on the left and fires a shot which is blocked by Toure.
41 minKagawa is brought down by Silva just ahead and to the left of the Monaco box, Reus aims for the top corner from the set piece but Subasic is behind it all the way.
45 minJust one minute added on at the end of the first half.
8.37pmHalf time here at the Stade Louis II and AS Monaco lead Borussia Dortmund 2-0 on the night, and 5-2 on aggregate. The visitors must score three unanswered goals in the second half just to keep the tie alive.
8.39pmThe first half hour or so of the game belonged entirely to Monaco, who broke the deadlock just three minutes in through Kylian Mbappe and doubled their lead in the 17th minute through Radamel Falcao. Thomas Tuchel opted for a 27th-minute substitution, bringing on Ousmane Dembele, and after his introduction, Dortmund have clawed their way back into the game in terms of possession and attacking intent - everything but score, so they still have a mountain to climb after the restart.
8.44pmHALF-TIME STATS: Monaco have had six shots (four on target), completed 108 passes, won two corners and enjoyed just 34 percent possession, while Dortmund have had four shots (two on target), completed 261 passes and enjoyed 66 percent possession, though they are yet to win a corner (or indeed score).
8.47pmA sliver of hope for Dortmund, who have finished the first half strongly - could a comeback be on the cards?
49 minOne down, two to go for Dortmund! Dembele is the key man here as he breaks down the right, skins Mendy and cuts the ball back into the middle for Reus to finish first time from 10 yards out!
51 minCHANCES! But here come Monaco as they very nearly restore their two-goal advantage on the night - Mbappe forces a corner which is swung in towards Jemerson, who nods towards goal but Burki leaps to put the ball behind again, and then Falcao gets a free header which goes just wide!
53 minWe have a break in play as Ginter takes a knock as he challenges for a header with Falcao, but after a bit of treatment he gets up and seems alright.
55 minA 3-2 win for Dortmund on the night is still a possibility, which is the only result that would see extra time.
56 minSchmelzer threads a pinpoint pass through to Reus, but Toure slides in and blocks the cross that follows as the ball ricochets off Reus and goes out for a Monaco goal kick.
57 minDembele continues to pull the strings for Dortmund as he embarks on a run, drifts in from the right and almost releases Aubameyang with a through-ball down the middle.
58 minAubameyang wins a corner off Jemerson down the right, which is swung in by Guerreiro and very nearly drops to Aubameyang at the far post, but he can't connect properly and the ball goes out for a goal kick.
60 minDortmund win a free kick as Falcao brings down Kagawa, but Reus's effort fails to get past the first man.
62 minBut here come Monaco as Lemar comes forward down the left and whips a cross in at the near post that Falcao gets to ahead of Sokratis, but the defender sticks out a boot to block his header and the ball ends up going wide under pressure.
65 minCHANCE! Oh my days! Falcao is so so close to adding a third for Monaco as Glik sends him clear down the inside-right channel, but the Colombian appears to have too much time to think about it and, rather than shooting first time, he cuts back inside and Dortmund's defence is able to get back and charge his effort down.
67 minCHANCE! And now at the other end, Dembele embarks on a slaloming run and tries to find Reus only for Jemerson to clear, and not long after Schmelzer drives forward from left-back and decides to have a pop, but Subasic claims.
68 minSUBSTITUTION: Radamel Falcao's shift is over, he comes off to be replaced by Nabil Dirar.
69 minCHANCE! Toure, from the right-hand side, slips a ball inside for Mbappe, who marauds into the box and sends a powerful low effort towards bottom left of goal, but Burki palms the shot around the post - and it's given as a goal kick!
72 minSUBSTITUTION: A final roll of the dice for Tuchel as Christian Pulisic is brought on in place of Raphael Guerreiro.
74 minCHANCE! And it's Dembele with yet another great move, a scooped pass over the top that Kagawa flicks on for Reus, only for the latter to blast the ball straight at Subasic.
75 minJust quarter of an hour left in this game - 15 minutes for Dortmund to try and claim two unanswered goals to keep this tie going.
77 minCHANCE! Lemar has a pop but his effort is charged down, then Silva's cross is turned behind by a defender for a corner which is swung in towards Bakayoko, who jumps and connects but his header goes over for a goal kick!
81 minSUBSTITUTION: Monaco make a switch as Valere Germain comes on for the superb Kylian Mbappe.
83 minSurely, surely that's a bridge too far for Dortmund now! Germain has been on the pitch for less than half a minute but still gets himself on the scoresheet, a simple sidefooting effort into the net after Lemar charges down the left on the break and plays a low cross into the centre for the sub to convert!
85 minNow Silva embarks on a solo run and makes headway deep into Dortmund's half, but Sokratis weighs in with an excellent tackle to thwart the danger!
87 minBarring some absolutely outrageous turns of events in both of tonight's CL fixtures, it's going to be Real Madrid, Atletico Madrid, AS Monaco and Juventus in the semi-finals.
88 minCHANCE! Pulisic turns in the box and attempts a snap-shot but it flies wide of target.
90 minSUBSTITUTION: One for Monaco as Andrea Raggi replaces Bernardo Silva, with two additional minutes to be played.
90+2 minFULL TIME: AS MONACO 3-1 BORUSSIA DORTMUND
9.40pmAnd that's it! Borussia Dortmund have battled valiantly, they have overcome adversity, but their Champions League adventure is over for the season as AS Monaco make it through to the semi-finals!
9.43pmThomas Tuchel's charges got off to a poor start as Kylian Mbappe and Radamel Falcao scored one apiece in the first 20 minutes of the second leg, and even though the visitors rallied round with Marco Reus's strike three minutes into the second half, Monaco weathered the storm and made sure of progress with Valere Germain's 81st-minute goal.
9.45pmSo AS Monaco join Real Madrid, Atletico Madrid and Juventus in the final four of the competition. The draw for the semi-finals will be held on Friday, April 21 at 11am BST, with the first legs being played on May 2 and 3, and the second legs on May 9 and 10. Thanks for following Sports Mole today and, until next time, goodbye!