The Belgian had played down expectations ahead of the match by claiming Zenit belonged in the Champions League and branding his side underdogs.
But his players proved otherwise as a Luuk de Jong brace either side of a Denny Landzaat effort put them firmly in the box seat ahead of next week's second leg in Russia.
At 2-0 down, the visitors might have still fancied their chances at home, but De Jong's second deep into injury-time all but killed off their hopes.
Twente were familiar with Russian opposition, having knocked out Rubin Kazan 4-2 on aggregate in the last round, and they appeared to have the measure of Zenit in the opening exchanges.
The Russian domestic season has yet to get under way and, although Zenit beat CSKA Moscow in the Russian Super Cup last weekend, they looked distinctly rusty tonight.
Both sides struggled to carve out clear-cut chances tonight until the hosts went ahead in the 29th minute.
Nacer Chadli sent in a cross from the right and De Jong guided a finish high into the net from the far side of the six-yard box.
Douglas then headed over from Theo Janssen's outswinging corner as the Dutch side looked to press home their superiority.
Zenit responded with several balls into the Twente danger area, but the hosts' defence dealt with them comfortably.
Bruno Alves had a shot stopped by Twente goalkeeper Nikolay Mihailov at the start of the second half before the hosts doubled their lead in the 56th minute.
Alves cleared Janssen's free-kick delivered from the left, but the danger was not over as Landzaat fired a shot across keeper Viacheslav Malafeev and into the bottom left corner from 18 yards.
An away goal for Zenit would have changed the complexion of the tie, but they rarely threatened the Twente backline, crosses or shots from distance which were easily dealt with all they could muster.
And instead it was the hosts who added a third in injury-time, De Jong heading Janssen's left-wing cross low into the net to leave the 2008 UEFA Cup winners with a mountain to climb to make the last eight.
FC Twente 3-0 Zenit St Petersburg: Match Report - view commentary, squad, and statictics of the game as it happened.