Graziano Pelle, who scored in the 3-0 first-leg win last week, fattened the Saints' already-healthy lead on four minutes and that proved enough to win on the night.
Sadio Mane then doubled the lead late on as Ronald Koeman's side cruised into the next phase of Europe's secondary competition.
The draw for the playoffs - the last round before the group stage commences - is scheduled to take place on Friday.
Below, relive the action with Sports Mole's live commentary.
Good evening! Are we ready for some Europa League action? Excellent. I'm Shane and I'll be taking you through all the action in Vitesse as Southampton look to reach the Europa League playoffs - and they most definitely should. Ronald Koeman's side only bring a 3-0 lead with them from a superb first-leg performance! Surely they'll reach the next round?
We don't want to jinx the visitors here, but elimination for Southampton tonight is about as likely as Koeman demanding a DNA test on the basis that he is Ed Sheeran's biological father. It's about as likely as Matt Le Tissier boycotting the 'banter' code on Soccer Saturday. It's about as likely as that Portsmouth fan - who quite literally changed his name to John Portsmouth - donning a red and white shirt tonight. In other words, a Vitesse comeback isn't too likely, my friends.
Goals from Shane Long, Dusan Tadic and Graziano Pelle have put Southampton on the brink of the playoffs. Okay it's not the Champions League - although at one stage last season it seemed like it could've been. It's the Europa League - its less famous and less sought-after sister; basically the Khloe Kardashian of football competitions. In any case, reaching the group stages would still represent a huge achievement given where the club was not too long ago.
It's easy to forget but Southampton went into administration and were relegated from the Championship back in 2009. Fast forward six years and they're very, very close to mixing it with some very stellar European teams it must be said. Fantastic stuff all round.
Shall we bring you some team news? Sure, but first let's provide a soundtrack to Vitesse's hopes of a comeback. One in a Million by Guns and Roses? No Hope by The Vaccines? Vince McMahon's entrance song? In fact, let's be optimistic:
Otay let's start with the visitors and I can tell you that Koeman names a very strong first XI, with three changes from last week's first leg. Steven Caulker, Cuco Martina and James Ward-Prowse come into the fold.
What do we make of Koeman's XI then? The Dutchman vowed to play a strong team and he has stuck to his word here. Risky or honourable, considering their Premier League season kicks off in three days? Hmm. Still, it proves that he is respecting the competition, so we can't complain about that.
Speaking of Pelle, how important is he for the Saints this season? You may remember that the Italian striker literally could not stop scoring goals early on in the 2014-15 campaign, before hitting a rough patch. He scored 16 goals in all competitions - but only five of those came in 2015! It sums up the severity of his Jekyl & Hyde syndrome last term and consistency is key for him this time around.
On Sunday, Southampton kick off their 2015-16 visit with a trip to Newcastle United and they'll need Pelle firing on all cylinders throughout the season in order to avoid being labelled a one-season wonder.
They were brilliant for most of last season and looked genuine contenders for a Champions League spot, but their campaign fizzled out a little. How many times have we seen teams flirt with the top four, before regressing in the following season? It's very common, and Pelle, along with Shane Long, Dusan Tadic and Sadio Mane, have a responsibility to help them kick on from their sixth-placed finish last time.
Incidentally, I expect Mane to have an excellent season. He found himself some stunning form toward the end of last season - and beat Robbie Fowler's record of the quickest hat-trick in Premier League history by scoring three against Aston Villa in two minutes and 56 seconds, which is ridiculous when you think about it.
Five minute warning: stick the kettle on now. Get the jaffa cakes. Pull the curtains. Send the wife an 'x' in a text message so you don't have to get up and kiss her when she gets home. We're not waiting for you. You've been warned.
Ignore that last post: there's going to be a slight delay. The players are only out on the field now but we still need to do the handshakes and all that jazz. Kickoff is in about two minutes I'd say.
CHANCE! Worryingly for Southampton, they've already conceded a chance. Kevin Diks races down the right hand before cutting back for Isaiah Brown, whose first-time hit is tame and it rolls wide. It goes down as a chance. No doubt about that.
Hmm, okay Vitesse now need FIVE goals in order to progress. It's a man we spoke about before the game, as the ball drops to Pelle, who expertly turns Diks in the box before lashing a left-footed strike across goalkeeper Room into the bottom corner. Fine goal that.
SHOT! Vitesse respond well to be fair as Rashica latches on to Brown's through ball before running at Caulker, but his shot was a little hurried and it bounces off the Saints defender for a corner, which then comes to nothing.
Vitesse well on top now but you have to imagine it's by design. 4-0 ahead on aggregate, Southampton don't need to expend much energy here - and especially not when the Premier League kicks off this weekend.
SHOT! What an effort this is from left-back Maikel van der Werff, who takes aim from well outside the box but his swerving effort is tipped over by Stekelenburg, and the corner again comes to nothing.
Despite trailing by four goals, Vitesse fans ensure there's a carnival atmosphere inside this massive stadium. In their yellow and blue shirts, they look like a bunch of bumblebees - but their team has no sting.
Interestingly, it appears as if Jon Dahl Tomasson - legendary Danish striker and former Newcastle United man - is Vitesse boss Peter Bosz's assistant. You learn something new every day.
CHANCE! Another decent chance for the hosts as Valeri Kazaishvili, in acres of space on the 18-yard line, goes for goal but his effort is straight at Stekelenburg. Should've done better there.
If you were expecting something of a goal marathon after Pelle opened the scoring within four minutes then you'll probably be disappointed because it's all Vitesse. Understandably, Southampton, who are 4-0 ahead on aggregate, aren't expending too much energy with attacking. They're sitting back, absorbing the pressure.
Brilliant defending from Caulker, as Rashica looks to turn the big centre-half en route to going through on goal, but the ex-Spurs defender snuffs out the danger. He had to, because Rashica was straight through with Stekelenburg there.
We're about 10 minutes from half time then and Saints still cruising. Well, not in the match, but in the tie. They're 4-0 ahead on aggregate, with Pelle scoring after four minutes to open up a 1-0 lead on the night.
CHANCE! It could, and should, be 5-0 on aggregate, as Sadio Mane knocks the ball over the head of Leerdam before being one-on-one with Room, but the winger bottles it and fires straight at the goalkeeper. Probably won't be too costly, will it?
Victor Wanyama goes to ground inside the box like a sack of spuds, but if you can picture Wanyama, it'd take a Hummer to take him off his feet. It was a dive to be honest and the referee is having none of it. Thankfully, he looked a little embarrassed, so that should be that as far as his diving career goes.
Two minutes until the break then we'll probably see a few Southampton changes at the break. I initially wrote 'quite a few' but remembered this isn't a friendly. It feels like one.
That's it for one half. Graziano Pelle's goal after four minutes has given Southampton a 4-0 aggregate lead in Arnhem. That should be that you suspect? Probably, considering Vitesse need five goals in the next 45 minutes, but let's not rule it out. Or definitely rule it out. Whatever.
RESTART! Aaaaaaaaaaaaand we're back underway in Arnhem. Vitesse, who trail 4-0 on aggregate, kick us off once again. Let's not beat around the bush: they need a miracle; one that would've made Jesus Christ himself say "how in the name of, well, me, did you do that?"
Pretty much same story different half for both these sides. Southampton have the deck chairs out effectively, whereas Vitesse know they're pretty much in a losing battle here.
It's surprising to see no subs from Southampton. Surely the match is done and dusted but instead of withdrawing his players in favour of preserving their fitness for Newcastle, Koeman has left them on. Strange.
UPDATE: I can tell you that West Ham are 2-1 down away to Astra Guirgiu - and 4-3 down overall. Another for the Hammers would level up the tie and should it end 2-2, we'd have extra-time. We can't have extra-time here, which is a good thing because both sets of players want this to be over I think.
SHOT! Mane races on to Pelle's over-the-top ball, before lashing a half-volley over the bar from a ridiculously tight angle. Scoring just wasn't an option there, but Mane can do what he wants - he's only the quickest scorer of a hat-trick in Premier League history and don't you forget it.
SAVE! Kazaishvili, who missed a good chance in the first half, misses another one here as he drives through Southampton's midfield before going for goal from outside the box, but Stekelenburg is down quick to deny him. Should've done better there.
SHOT! This is good from Marvelous Nakamba, who nips in ahead of Ward-Prowse before firing a fierce effort over the crossbar from just outside the area. What a fabulous name, though. Ward-Prowse. Genius.
Vitesse striker Rashica is a good player. I fancy him to go on to bigger and better things at some point in the future. He has caused Southampton's defence plenty of problems tonight.68
Not going to lie: the most interesting thing in this half was hearing Ronald Koeman use the F word. The bad F word; the one that rhymes with book and not beck. Other than that, it's been a very disappointing and boring half.
BOOKING: Stekelenburg goes into the book for time wasting it seems. Pretty needless yellow card it has to be said. Time wasting. What are they going to do, score five times in the last 15 minutes? No need pal.
UPDATE: West Ham look set to exit the Europa League before the playoffs as they trail Astra Guirgiu 2-1 on the night and 4-3 on aggregate. One more away goal for the Hammers would take that into ET.
That's that. Sadio Mane, who got two assists last week, gets a goal for his efforts as he drives his effort beneath the body of Room to make it 5-0 on aggregate.
And that's all she wrote in Arnhem. Southampton advance into the playoff stages as we thought they would. Pelle's fourth-minute opener is enough to give Saints a 1-0 win on the night - and 4-0 on aggregate. They never looked in real trouble, to be fair. Vitesse had one or two chances, but nothing that was ever going to turn a tie that effectively ended when Koeman's men won 3-0 last Thursday. Anywho, that's all from us. Hopefully you've enjoyed our coverage and we'll see you next time.
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