Pelle opened the scoring with a vicious first-half strike, before Tadic doubled his side's lead from the penalty spot after Sadio Mane was fouled in the area.
Long then completed the rout with a simple header in the dying stages following a superb Juanmi cross.
The result gives Koeman's team a comfortable lead to defend when they head to Arnhem for the second leg, having displayed admirable fitness and composure in their first competitive game of the season.
The hosts improved as the game wore on, having shown initial signs of nerves while Vitesse created the better of the early openings.
Georgian striker Valeri Kazaishvili gave Southampton a wake-up call when he latched onto a long ball and managed to get away from his marker before dragging a shot just wide of Maarten Stekelenburg's post.
As the home side settled, Maya Yoshida should have done better from a Steven Davis corner, and an even better chance followed when Victor Wanyama found himself with time and space inside the box, but the Kenyan's aim was poor and Vitesse goalkeeper Eloy Room made a relatively simple save.
The Premier League side found their rhythm by the half-hour mark, and Davis stung Room's hands when he arrowed a fierce shot from just outside the box.
Southampton's dominance paid off when Cedric broke down the right flank and fed Mane, whose simple pass into the box left Pelle with only Room to beat, and he did so with an instinctive shot that flashed into the top of the net.
On the cusp of half-time, the hosts doubled their lead, as Mane was brought down in the box by Rochdi Achenteh, and Tadic made no mistake with the spot kick.
On-loan Stekelenburg, making his Southampton debut, enjoyed a quiet game until Danilo Pantic broke through after 70 minutes, but the Dutchman was alert to the task, saving low to his right.
Vitesse substitute Milot Rashica missed a clear chance, sending a volley over the bar from 10 yards out when a more accurate shot might have put his side right back in the tie.
Moments later, second-half substitute Long broke clear of the defence and was denied by Room, but fellow replacement Juanmi flicked the loose ball back to the Irishman and he made no mistake with a close-range header.
