Ronald Koeman's side have been playing their initial European match due to the fact 2002/03, and Pelle got the Saints off to a perfect start midway through the very first half when he powered property.
The ball was threaded by means of to the Italian forward by Sadio Mane, and Pelle did not hold back with his finish, rifling the ball higher into the leading right-hand corner on the net.
Southampton had been heavy favourites heading in to the tie, and that proved to become correct when Dusan Tadic added a second aim in the penalty spot with all the last kick with the initial half.
Mane was once again involved as he turned his man on the byline, ahead of getting brought down in the box to present his side a golden possibility in the spot. Tadic stepped up and produced no mistake as he slotted low to the keeper's suitable.
Within the second half, Saints continued exactly where they left off and only had one brief scare inside the 71st minute when Maarten Stekelenburg was forced to save properly from Vitesse substitute Milot Rashica.
Because the match entered the final 10 minutes, Saints created certain of their comfortable victory by adding a third. Chelsea loanee Lewis Baker's sloppy mistake around the half-way line enabled Lengthy to nip in and steal the ball, just before sprinting towards the Vitesse box.
One-on-one with the goalkeeper, he couldn't slot his initial effort house, the ball instead ricocheting out to fellow substitute Juanmi on the right-hand side.
The Southampton new boy developed a moment of magic to take the ball towards the byline, before flicking a deft chip across to Extended inside the centre on the six-yard box, who couldn't miss with his free header.
Koeman's guys now appear forward to a trip to Holland next Thursday, where their 3-0 lead really should be sufficient to safe them a spot within the play-off round of your Europa League.