Ronald Koeman made three changes for the trip to Arnhem, with Cuco Martina and Steven Caulker handed their competitive debuts, while James Ward-Prowse filled in for the injured Jordy Clasie.
Saints would've been dreaming of an early goal to kill the tie, and their wishes came true when Pellè powered a low left-footed effort into the corner of the Vitesse net after just four minutes of play.
Ward-Prowse assisted the Italian, after poking Dušan Tadić's cross into the path of the one-time Feyenoord man, who promptly swivelled before chalking up his sixth goal in his last six games against Vitesse.
Vitesse looked to hit back quickly, with first Maikel van der Werff and then Valeri Kazaishvili trying their luck against Saints stopper Maarten Stekelenburg. Van der Werff unleashed a thunderbolt from range that was tipped over by the arm of Stekelenburg, before Kazaishvili's powerful attempt from the edge of the 18-yard-box was caught by the former Ajax keeper.
Sadio Mané then saw his low effort save by Vitesse goalkeeper Eloy Room, despite the Senegalese forward losing his marker and venturing into the penalty area.
Cuco Martina saw himself yellow carded for a poor tackle on Chelsea loanee Lewis Baker, before Greek match official Ilias Spathas called time on the first period of the fixture.
Pellè started the second half by scooping the ball over the Vitesse defence, but Mané was unable to capitalise, instead firing wide of the target.
Vitesse went in pursuit once again, as they chased their first goal in six games against English opposition. Kazaishvili worked some space, but his low drive was countered by a strong diving save that saw Stekelenburg clutch the ball.
Juanmi then replaced Tadić in the 65th minute, with Vitesse boss Peter Bosz bringing on Nathan in place of Isaiah Brown.
Goalscorer Pellè then departed in the 72nd minute, as Koeman gave Jay Rodriguez his first competitive appearance since picking up a serious knee injury at Manchester City's Etihad Stadium in April 2014.
Rochdi Achenteh then came on for Vitesse, with Van der Werff making way.
Southampton's final change came in the 77th minute, with Harrison Reed replacing Steven Davis.
Four minutes later, the hosts made their final change, sending on Danilo Pantić to replace Baker.
Pantić proved to be at fault for Southampton's second, as he was dispossessed by Mané, who then charged at goal before stroking an effort past Room and into the back of the net to make it 5-0 on aggregate.
Victor Wanyama almost made it 3-0, but his header was saved by Room after an inviting free-kick delivery from Ward-Prowse.
Saints will learn of their opponents for the final round of qualifying in the Europa League play-off draw, which is made at 12:00 BST on Friday.
Southampton: Stekelenburg; Martina, Fonte, Caulker, Yoshida; Ward-Prowse, Wanyama, S. Davis (Reed - 77'); Mané, Pellè (Rodriguez - 72'), Tadić (Juanmi - 65').
Unused Substitutes: K. Davis, Cédric, Targett, McQueen.