Twente have 34 points from 15 matches, one more than PSV and three ahead of third-placed Vitesse Arnhem who host Roda JC Kerkrade on Sunday.
Den Haag's Tjaronn Chery was sent off in the 49th minute and the home team quickly took advantage as Dmitry Bulykin set up Dusan Tadic for a goal six minutes later.
Russian Bulykin then made it 2-0 with seven minutes to go to earn coach Steve McClaren's team their first victory since November 4.
Ajax, playing in front of a 51,064 crowd in Amsterdam, were rewarded for a bright start when Siem de Jong headed in after 29 minutes.
Jeremain Lens equalised for PSV seven minutes later with a curling, angled shot that beat keeper Kenneth Vermeer.
PSV missed good chances to go in front through Lens and Tim Matavz before Danny Hoesen restored the lead for Ajax with a near-post finish following excellent work from Viktor Fischer who beat three opponents in the 72nd minute.
Fischer made sure of victory with three minutes to go, pouncing on a poor header from Dries Mertens to hand PSV their second straight defeat.
"That was a fully-deserved win because we applied very aggressive pressure and never allowed PSV to play their game," Ajax coach Frank de Boer told reporters.
De Boer's side are fourth on 30 points, ahead on goal difference of Feyenoord who beat RKC Waalwijk 2-0 in the battle between coaching brothers Ronald and Erwin Koeman.
Italian Graziano Pelle headed the opener for Ronald's men on the stroke of half-time before adding the second 12 minutes into the second period, controlling the ball on his chest and volleying home to lift his league tally this season to 10 goals.