Twente were quick out of the blocks at De Grolsch Veste on Wednesday and took the lead in the ninth minute through Felipe Gutierrez.
However, Twente had Rasmus Bengtsson sent off in the 64th minute when he brought down Michiel Kramer, who was through on goal.
Luc Castaignos then had a goal ruled out for offside as the Enschede outfit looked to double their lead, before Kramer headed Den Haag level.
Twente were reduced to nine before full-time, as Castaignos was dismissed for a reckless challenge on Dion Malone.
The result effectively rules Twente out of the title race, leaving them 12 points behind leaders Ajax, with only 12 points left to play for.
The point does, however, lift them above Vitesse, whose links with Chelsea were revealed to be under investigation by the Dutch Football Association on Wednesday.
Two first-half goals from Joel Veltman and Davy Klaassen were enough to keep Ajax on course for the title with a 2-0 win at RKC Waalwijk, who remain in the bottom three.
The Amsterdam giants can now win the league this weekend if they win and Feyenoord fail to beat RKC.
Feyenoord kept up their slim hopes as they triumphed 2-1 over 10-man Heracles on Wednesday.
Iliass Bel Hassani gave Heracles a 14th-minute lead, before Mitchell te Vrede and Tonny Vilhena gave Feyenoord the lead heading into the interval.
Simon Cziommer's dismissal for a second bookable offence in the 87th minute halted Heracles' chances of a comeback.
Go Ahead Eagles climbed four points clear of the relegation play-offs courtesy of an own goal from Henrico Drost in the second half of a 2-1 win over NAC Breda.
In the day's remaining fixture, Utrecht pulled themselves level on points with opponents Cambuur with a 1-0 home win, lifting them five points clear of the danger zone.
Jacob Mulenga scored the winner, with Cambuur's hopes of a comeback dented midway through the second half when Lucas Bijker was sent off.
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