The Spanish side held a commanding 5-1 first-leg lead but Emir Bajrami's first-half goal raised slim hopes for the hosts, needing to win 4-0 at De Grolsch Veste.
They were unable to build on that, though, and Giuseppe Rossi's penalty - and a red card for Dwight Tiendalli - ended even the slightest doubts before a late second spot-kick from Marco Ruben and a fine finish from Cani rounded off an 8-2 aggregate win for the Yellow Submarine.
The heavy first-leg deficit meant Twente needed goals desperately but Bajrami's ferocious long-range effort, blocked by goalkeeper Diego Lopez, was the only shot in anger in a lacklustre opening 20 minutes.
Villarreal full-back Mario Gaspar shot just wide of the near post when well placed at the other end but chances were at a premium.
Twente broke the deadlock, though, when Bryan Ruiz slipped the ball to Theo Janssen, who produced a superb back-heeled return pass to play the striker into the penalty area.
As Lopez came out to narrow the angle, Ruiz squared unselfishly for Bajrami to slide in and tap the ball into the empty net while team-mate Marc Janko appealed justifiably but needlessly for a penalty, having been hauled down by Mateo Musacchio.
More superb work from Bajrami almost led to a second goal. The tricky playmaker twisted through defenders and fizzed a low cross across the edge of the six-yard box but it was too far in front of Ruiz and even the towering Janko, as both lunged despairingly at the ball.
Twente keeper Sander Boschker punched clear Borja Valero's corner and was back in position to gather Mubarak Wakaso's looping header to preserve his side's lead on the night into half-time.
Rossi blazed a free-kick high and wide after the resumption as he sought to catch Boschker off guard and the keeper then saved from the Italian after a muscular jinking run.
It was Rossi who killed the tie off once and for all, though, from the penalty spot.
Tiendalli's appalling touch in his own penalty area was seized on by Mubarak and Tiendalli brought the Ghanaian youngster down and was sent off. Rossi tucked away the spot-kick, sending Boschker the wrong way after a staccato run-up.
Musacchio's unchallenged surge into attack ended with the veteran Boschker unconventionally saving from Ruben with his face and, with Twente seemingly a spent force, Ruben was sufficiently emboldened to try his luck from 40 yards. Boschker saved with ease.
Hernan Perez carved out a chance to mark his European debut with a goal but, after stepping neatly inside Thilo Leugers, he flashed a shot wide - an effort soon emulated by Ruben.
Perez's lightning pace drew another foul in the penalty area though, this time from Bart Buysse, and Ruben finished low beyond the dive of Boschker.
And Cani rubbed salt into the wounds, lashing home after 40-year-old Boschker had saved from Ruben. Boschker was lucky to see a stoppage-time clearance bounce wide off Ruben and Alexander Bannink failed to convert Bajrami's cross at the other end.
FC Twente 1-3 Villarreal: Match Report - view commentary, squad, and statictics of the game as it happened.