The hosts took leads through first-half goals from Marko Vejinovic and Jan-Arie van der Heijden.
However, Jesus Corona's strike kept Twente in contention and they were given a chance to equalise from the spot as Davy Propper was dismissed for halting a fine counter-attack in the penalty area, and Hakim Ziyech netted from 12 yards.
Though enjoying a man advantage and the momentum, Twente could not find a winner and were unable to climb into the top four - the point keeping them sixth.
PEC Zwolle maintained their impressive home record this season, seeing off ADO Den Haag with a 3-1 success.
Ron Jans' side prevailed in front of their own fans for the sixth time from eight games this term thanks to a Tomas Necid double and Jesper Drost's strike, with Michiel Kramer grabbing a consolation for the away team.
Heracles continued their recent resurgence when completing a hat-trick of wins by prevailing 4-2 on the road at Utrecht.
After starting the campaign with seven straight losses, they have now come out on top in five of their last eight matches.
Thomas Bruns, Mark-Jan Fledderus, Wout Weghorst and Iliass Bel Hassani did the damage, with Utrecht only able to reply through Kristoffer Peterson and Gevero Markiet.
Groningen and Herenveen drew 1-1 at Euroborg, with Michael De Leeuw putting the hosts ahead but Mark Uth levelling.
