Jens Toornstra's 32nd-minute strike put Utrecht ahead at De Grolsch Veste before Twente defender Rasmus Bengtsson saw red a minute before the break.
Jacob Mulenga scored from the spot on the stroke of half-time and although Dusan Tadic reduced the arrears after 56 minutes, Dave Bulthuis soon restored the visitors' two-goal lead.
Tadic scored from the spot with 16 minutes left, but Mike van der Hoorn found the target for Utrecht to leave Steve McClaren's side four points off leaders PSV.
Second-placed Ajax are three points off top spot following a comfortable 3-0 win at VVV Venlo.
Striker Siem De Jong set the ball rolling in the 34th minute, before second-half strikes from Viktor Fischer and substitute Derk Boerrigter sealed the win.
Feyenoord in fourth are five points off top spot after they won 3-1 at Willem II.
Graziano Pelle gave the visitors a 10th-minute lead before Virgil Misidjan equalised from the penalty spot seven minutes before the break. Tony Vilhena scored a second-half brace for Feyenoord to ensure victory.
Elsewhere, NEC Nijmegen beat Vitesse Arnhem 2-1 thanks to goals from Remy Amieux in the 38th minute and Ruud Boymans in the 67th, with Mike Havenaar on target for Vitesse.