With Ronald Koeman confined to his seat on the bench after injuring himself training with his players, the Saints rarely gave him cause for concern, as Graziano Pelle’s neat turn and finish settled any early nerves in the GelreDome.
The rest of the match was a mere procession, as Saints sat back and soaked up pressure from the Eredivisie side, before Saido Mane made sure of the win late on.
The good news kept coming for the travelling Saints fan, despite their clashes with police earlier in the day, with Jay Rodriguez returning to competitive action for the first time in 16 months.
Reports of crowd trouble outside the stadium failed to unsettle the Saints, with Koeman naming a strong side despite the 3-0 scoreline from the first leg.
And after just four minutes, the Premier League side put the tie beyond doubt, as Pelle, who has a great record against Vitesse, picked up the loose ball from James Ward-Prowse, swivelled and drilled the ball into the bottom corner.
Vitesse’s side boasted two Chelsea loanees, Izzy Brown and Lewis Baker, but neither could influence proceedings.
Mane was lively in the first 45 minutes, and could have doubled the visitors lead, but scuffed an effort from a good position straight at the goalkeeper.
It was more of the same after the break, with Saints happy to take their foot off the gas.
Rodriguez’s introduction, replacing Pelle, brought a rapturous applause from the Saints contingent, and he looked lively in his first outing since rupturing knee ligaments at Manchester City 16 months ago.
The former Burnley hitman almost made an instant impact, firing a dipping and swerving long-range effort at goal, but it flew just over.
However, Southampton did put the icing on the cake late on as Mane capitalised on an error from another Chelsea loanee Danilo Pantic to race clear and slot home to net a deserved second.