Alkmaar, trailing to Belgium international Nacer Chadli's goal on 20 minutes, had Jozy Altidore and Adam Maher dismissed within a minute early in the second half.
The match flared up when the visitors were denied a penalty after a foul on Maher.
Twente compounded Alkmaar's woes by adding to their lead through Dusan Tadic's penalty and a Luc Castaignos strike.
Steve McClaren's side have 21 points after eight matches, three ahead of PSV Eindhoven who cruised to a 4-0 home win over NAC Breda, with Jeremain Lens opening the scoring before delivering three assists.
Feyenoord remain fifth with 14 points after salvaging a point in a 2-2 draw at Groningen thanks to Bruno Martins Indi's last-minute goal.
Ronald Koeman's team played for more than an hour with 10 men after teenage defender Terence Kongolo was sent off.
