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.
Midfielder Maher was shown a straight red card for kicking a Twente player, then U.S. international Altidore got his marching orders for dissent.
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.
Vitesse Arnhem, who drew 3-3 against Heerenveen on Saturday, are third on 18 points, above Ajax Amsterdam who slipped to fourth after being held to 1-1 draw by Utrecht.
Ryan Babel finished a well-executed attack after six minutes to put Ajax ahead but early in the second half Fabian Sporkslede headed into his own goal.
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.
- Adam Maher
- Jozy Altidore