A Masimba Mambare header, which was the team's first goal of the tournament, was enough for the Warriors to take the win.
The other result in Group B means that Zimbabwe will face Mali in the quarter-finals.
Read how all the action unfolded below:
4.32pmGood afternoon! It's around half an hour until Burkina Faso and Zimbabwe get their final Group B game underway, with a place in the quarter-finals still up for grabs. Let's start with a look at the teams...
4.34pmBURKINA FASO STARTING XI: M Kabore, Sango, Dayo, Kone, Gouo, To, Zoungrana, S Traore, Y Kabore, E Traore, Ouedraogo
4.36pmZIMBABWE STARTING XI: Chigova, Zvirekwi, Jaure, Chipeta, Ncube, Mambare, Phiri, Moyo, Mahachi, Sibanda, Sithole
4.38pmBURKINA FASO SUBSTITUTES: Diarra, Kobre, Coulibaly, Yeye, Nebie, Bayala, Nimi, Snaou, Barro, Tiendrebeogo, F Kabore, Simpore
4.40pmZIMBABWE SUBSTITUTES: Diya, Dube, Muparati, Mbara, Mazivisa, Machapa, Ngomo, Sadiki, Chindungwe, Manganga, Ndlovu
4.43pmBassirou Ouedraogo keeps his place up front for Burkina Faso, having scored the equaliser that has kept them in the tournament in their previous match against Morocco.
4.46pmZimbabwe are still looking for their first goal of the tournament, having played to goalless draws with Uganda and Morocco in their previous matches.
4.50pmThe other two nations in Group B are also due to kickoff at 5pm. You can follow live coverage of Morocco vs. Uganda as well with Sports Mole this afternoon.
4.54pmPREDICTION: Before the action gets underway, it's time for a quick prediction. Burkina Faso may have fewer points of the board, but they've looked the more dangerous in front of goal so far in this tournament. I think they can edge this 1-0 this afternoon.
4.57pmHere come the teams. Naturally, the anthems of both countries will be played prior to kickoff, and then we'll be underway.
1 minBurkina Faso get the match underway.
3 minBurkina Faso are on the attack from the start, but struggle to penetrate a Zimbabwe defence that it yet to concede a goal in this tournament. Eric Traore eventually unleashes a powerful strike from outside the box, but it's easily blocked.
5 minZimbabwe get their first break of the game, but Mahachi is brought down before he can get the shot away. However, it does give Zimbabwe a free kick from a dangerous position.
7 minCHANCE! Moyo's delivery is accurate and has plenty of power behind it, but Mohamed Kabore makes a superb save in the Burkina Faso goal, tipping it over the crossbar, before his teammates quickly clear the resulting corner. This has been a bright start by the side that finished as runners-up in the Africa Cup of Nations last year.
10 minSimba Sithole wins a free kick out wide for Zimbabwe following a lunging challenge from Mbah Kone. The ball is whipped into the box, but Kabore comes off his line to punch it away, before eventually collecting it when it's sent back in. The Burkina Faso keeper has had a bit to do in the opening stages of this match, but he's looking pretty sharp so far.
13 minCHANCE! Zimbabwe have been starting to dominate the possession, and it results in another chance for them to score their opening goal of the tournament, but Peter Moyo's driven effort from just outside the box climbs too high, ending up in the crowd.
16 minSango clashes with Ncube on the edge of the Burkina Faso penalty area, giving Zimbabwe another free kick. Burkina Faso's tackling has been a little loose in the opening quarter of an hour.
18 minCHANCE! To turns the ball wide for a corner from the free kick. After a scramble in the box, it's eventually cleared as far as Mahachi, who unleashes a rocket from 25 yards, but Kabore makes another solid save to deny the midfielder.
20 minA long ball into the Zimbabwe box almost drops to Ouedraogo, but Chigova manages to jump up high and take the catch. However, the keeper collides with one of his defenders, leaving both of them needing treatment.
22 minAn ice pack on the knee appears to have solved the problem for Chigova, and it looks like Jaure is able to continue as well.
24 minCHANCE! Burkina Faso almost catch Zimbabwe out on the break, with Traore sending in a well-placed cross, but Jaure is able to deflect it behind, giving their opponents their first corner of the game.
25 minCHANCE! From the corner, it falls to Ouedraogo, but the goalscorer from the previous game smashes his finish just wide.
28 minThe game is really open at the moment, with chances coming at both ends of the pitch thick and fast. Zimbabwe win another corner, but Mahachi steps behind the goalline while trying to control it, giving Burkina Faso the goal kick. The Stallions quickly press down the other end, but a Traore cross has too much pace for Ouedraogo.
29 minUPDATE! Elsewhere in the group, Morocco have taken the lead against Uganda. If this game were to finish in a draw, both Burkina Faso and Zimbabwe would be out with a Morocco win.
31 minIt looks like George Chigova may have to go off, as he's sat down inside the Zimbabwe box rubbing his knee. He received treatment on it earlier, but it looks like that hasn't been enough. The physio is back on the pitch, but the stretcher it ready too.
34 minThe Zimbabwe keeper has plenty of tape applied to to affected knee and it looks like he'll be trying to carry on. He's yet to concede a goal in this tournament, and will surely be keen to see out the group stage with a 100% record.
36 minBOOKING! Yelleboure Alassane Sango is the first player to pick up a yellow card for a two-footed challenge on Mahachi. It's his second yellow of the tournament, so he'll miss the next game if Burkina Faso progress.
39 minBurkina Faso are finishing the half strongly, and have a free kick from a strong position, as Issouf Kabore is brought down just outside the box.
42 minCHANCE! Sango hits it brilliantly, but Chigova makes a stunning save in the Zimbabwe goal to keep it out, before the rebound is turned behind for a corner.
43 minZimbabwe try to get the ball back down the other end, but a loose ball in the box is swiftly claimed by Kabore.
45 minBOOKING! Masimba Mabare earns himself a yellow card for what was frankly a rugby tackle, dragging Sango down by the waist.
45+1 minWith the injuries earlier on, there are three minutes of stoppage time in this first half.
45+3 minZimbabwe are camped out in the Burkina Faso penalty area in the closing stages of this half, but can't get the breakthrough with plenty of men back for Burkina Faso.
45+4 minHALF TIME: Burkina Faso 0-0 Zimbabwe
5.52pmSo no goals in the first half, but both sides have been playing positively, and it certainly feels like there are goals in this game.
5.55pmBoth sides need to find a goal if they are to stay in the tournament. With Morocco beating Uganda, they would both be knocked out if the match finished goalless.
5.59pmThe two teams will need to keep an eye on their discipline in the second half. Both had a player booked in the first period, and there were certainly a few more challenges that might have earned a booking from another referee as well.
6.02pmThe players are making their way back out for the second half.
46 minZimbabwe get the action back underway.
48 minBurkina Faso are looking to create an early chance, with Traore playing his way into the box. Zvirekwi manages to clear the initial danger, but Kabore floats an effort back in, only for Chigova to claim the ball.
49 minBurkina Faso may be forced into a change early in this second half as Eric Traore is down injured. He would be a big loss for the Stallions as he was one of their most explosive players in the first half. He's being put on the stretcher, so it doesn't look like he'll be continuing.
50 minSUB! Traore does indeed make way, with Oumarou Nebie coming off the bench in the first change by either side in this match.
51 minCHANCE! Zimbabwe win a corner, and it falls nicely for Eric Chipeta, who tries to head it into the bottom corner, but it goes just wide.
53 minBOOKING! Elisee To is the first player to be shown a yellow card in the second half, charging into Peter Moyo in midfield.
54 minCHANCE! Following the booking, Zimbabwe quickly get the ball forward, with Mahachi picking it up inside the box, but he puts his shot wide of the left post.
56 minCHANCE! What a lively start it's been to the second half! Burkina Faso immediately put Chigova under pressure on the counter-attack, bringing the keeper off his line. He's left stranded as Kabore pulls it back for Ouedraogo, but Ncube is able to make the block.
57 minGOAL! Burkina Faso 0-1 Zimbabwe (Mambare)
57 minZimbabwe have their first goal of the tournament in Masimbi Mambare's first game of the group stage. Peter Moyo draws Chigova off his line with a superb cross, but the physcial midfielder manages to out-muscle the keeper, header it over his head to break the deadlock. If Zimbabwe can hold on, they will progress to the quarter-finals at the expense of Uganda.
59 minSUB! Simba Sithole makes way for the similarly-named Simba Nhivi Sithole.
61 minCHANCE! Zimbabwe are all over the Burkina Faso goal at the moment as they look to seal the win. Kabore tries to punch the danger away, but he mi-times it, and almost gives Sithole a free header into an empty goal, but the sub rushes it and puts it over with his first touch of the game.
62 minThat could be a very brief cameo from Sithole. He tries to find his way through for a second chance, but is brought down on the edge of the box, and has to go off for treatment, just three minutes after coming on.
65 minUPDATE! Uganda have equalised against Morocco in the other game in Group B. As Zimbabwe were level with Morocco on points coming into this match, the score doesn't affect Zimababwe's progress.
68 minBurkina Faso have had little in the way of chances since going a goal down. Zimbabwe are comfortably holding the ball in midfield before Mabare makes a superb run down the wing before getting the cross in, but Kabore is able to claim it.
70 minBurkina Faso finally get the ball back down the other end, but Zvirekwi blocks the through ball to put an end the move.
72 minSUB! Zimbabwe make their second change of the game, with Charles Sibanda heading off and Tawanda Muparati taking his place. It's a change that sees a more defensively-minded player replacing the striker, so Zimbabwe may be looking to protect the lead that they have rather than extend it.
73 minSithole, who seems to have shaken off his earlier knock, breaking superbly from midfield, dashing down the wing before moving into the box and eventually winning a corner.
74 minSUB! It's another change for Burkina Faso, as Elisee To is replaces by Aliassou Sanou.
76 minCHANCE! Burkina Faso concede yet another free kick. Phiti tries his look from over 35 yards out, and it's on target, but Kabore just manages to hold on to the well-struck effort.
78 minCHANCE! Burkina Faso have their first good chance since going a goal down. Sidiki Aboubacar Traore picks up the ball on the edge of the box, and takes a step or two further before trying to lob it beyond Chogiva, but the Zimbabwe keeper manager to hold on to it with a diving save. However, the save appears to have aggravated his knee injury from the first half.
80 minSUB! The shot will be Traore's last contribution, as he's heading off, with Ilias Tiendrebeogo replacing him for Burkina Faso.
81 minCHANCE! Peter Moyo hit the ball on the half-volley from just outside the penalty area, but it curls just wide at the last moment. Zimbabwe seems to have upped their intensity since the Traore chance, realising that the win isn't guaranteed yet.
82 minUPDATE! Morocco have regained the lead in their game against Uganda, which means that they will finish top of Group B is both games finish like this, with Zimbabwe progressing as runners-up.
84 minZimbabwe seem keen to try and keep the play in midfield, knowing that they will be through to the last eight if they keep Burkina Faso away from their goal. However, there's a brief stoppage as Moyo loses one of his boots while making a tackle.
85 minZimbabwe are starting to take up a more defensive formation, with all but three of their player inside their own half and behind the ball. However, Moyo still tries to give the side another chance blasting the ball forward with a long ball
86 minBOOKING! Issoufou Dayo picks up a late booking for a clumsy challenge on the goalscorer, Mambare, who needs to go off the pitch as a result.
88 minKabore puts a high ball into the box for Burkina Faso, and Chigova almost make a hash of his catch, but he manages to claim it at the second time of asking to make sure Burkina Faso don't get a chance to equalise.
90 minOFFSIDE! Zimbabwe try a late move to put the result beyond doubt, but Moyo is caught offside while making a run into the box.
90+1 minBurkina Faso have four minutes of stoppage time to try and find an equaliser, although that wouldn't be enough to see them through.
90+2 minBOOKING! Milton Ncube is the latest player to pick up a booking for time wasting.
90+2 minCHANCE! Ouedraogo misses the target by just a couple of inches as he tries to head in for Burkina Faso.
90+3 minSUB! Zimbabwe make their final change of the match, bringing on Oscar Machapa for Peter Moyo.
90+4 minCHANCE! Mahachi comes close to adding a second for Zimbabwe, attempting to slice it in from a tight angle out wide, but Kabore makes a solid save in the Burkina Faso goal.
90+5 minFULL TIME: Burkina Faso 0-1 Zimbabwe
6.54pmZimbabwe progress to the quarter-finals of the African Nations Championship with a 1-0 win over Burkina Faso, scoring their opening goal of the tournament.
6.56pmUPDATE! Morocco have just added a third against Uganda deep in stoppage time, which will be enough for them to top Group B.
6.57pmThe results mean than Zimbabwe will face Group A winner Mali in the next round.
6.58pmThat's all from this match, but join us again tomorrow when the Group C teams play their final games of the tournament, with Congo, Ghana and Libya fighting it out over two places in the last eight. Until then, thanks for joining me this afternoon/evening, goodbye!