Nacer Chadli opened the scoring after 20 minutes and AZ shot itself in the foot twice in consecutive minutes early in the second half.
The visitor had Adam Maher sent off in the 51st minute and Jozy Altidore dismissed a minute later and after that it was just a question of how many Twente would get.
Dusan Tadic doubled the advantage with a 60th-minute penalty and Luc Castaignos added a third two minutes later.
PSV Eindhoven moved level on points with Vitesse Arnhem in second with a 4-0 win at home to NAC Breda.
Jeremain Lens put PSV in front in the 24th minute and Ola Toivonen got a second on the stroke of half-time.
Mark van Bommel grabbed number three 11 minutes after the break and Georgino Wijnaldum completed the rout five minutes from time.
Champion Ajax slipped up as it was held 1-1 at home to FC Utrecht to fall five points off the early pace.
Ryan Babel put the Amsterdam side ahead in the sixth minute, but Fabian Sporkslede's own goal just five minutes after coming on ensured a share of the spoils.
Bruno Martins Indi scored a 90th-minute leveller to rescue a 2-2 drew for Feyenoord at FC Groningen.
Sekou Cisse put the visitor ahead in the ninth minute, but goals from Leandro Bacuna and Virgil van Dijk had Groningen ahead at the break.