Ronald Koeman’s side were playing their first European match since 2002/03, and Pelle got the Saints off to a perfect start midway through the first half when he powered home.
The ball was threaded through to the Italian forward by Sadio Mane, and Pelle did not hold back with his finish, rifling the ball high into the top right-hand corner of the net.
Southampton were heavy favourites heading into the tie, and that proved to be true when Dusan Tadic added a second goal from the penalty spot with the last kick of the first half.
Mane was again involved as he turned his man on the byline, before being brought down in the box to gift his side a golden chance from the spot. Tadic stepped up and made no mistake as he slotted low to the keeper’s right.
In the second half, Saints continued where they left off and only had one brief scare in the 71st minute when Maarten Stekelenburg was forced to save well from Vitesse substitute Milot Rashica.
As the match entered the final 10 minutes, Saints made sure of their comfortable victory by adding a third. Chelsea loanee Lewis Baker’s sloppy mistake on the half-way line enabled Long to nip in and steal the ball, before sprinting towards the Vitesse box.
One-on-one with the goalkeeper, he could not slot his initial effort home, the ball instead ricocheting out to fellow substitute Juanmi on the right-hand side.
The Southampton new boy produced a moment of magic to take the ball to the byline, before flicking a deft chip across to Long in the centre of the six-yard box, who couldn’t miss with his free header.
Koeman’s men now look forward to a trip to Holland next Thursday, where their 3-0 lead should be enough to secure them a place in the play-off round of the Europa League.