The defending champions, who are aiming to win the Dutch top-flight crown for a fifth-successive season, had the chance to go top of the table with a win ahead of PSV's clash with Feyenoord on Tuesday.
However, Frank de Boer's men could not produce the goods at the Kyocera Stadium, despite midfielder Davy Klaassen putting the visitors ahead after eight minutes by meeting Arkadiusz Milik's header back across goal with a fierce volley into the net.
Most would have expected Ajax to build on that lead and win comfortably, but the hosts refused to lie down and claimed a share of the spoils through Michiel Kramer's 10th goal of the season.
Den Haag right-back Gianni Zuiverloon was then sent off for a second bookable offence in the fifth minute of injury time, however, that dismissal came too late for Ajax to take advantage.
Twente missed the chance to gain ground on the top two and dropped below Feyenoord into fourth courtesy of a 2-1 home defeat to Groningen.
Alfred Schreuder's hosts went into the game on the back of three straight league wins and with an unbeaten home record that stretched back to May of last year, yet they fell two goals behind inside 27 minutes.
First Michael de Leeuw's diving header put Groningen in front before a tame free-kick from Tjarron Chery crept under Twente goalkeeper Nick Marsman.
The game looked to have turned when Groningen's Danny Hoesen was shown a straight red card for a foul on Cuco Martina nine minutes from time.
However, although Luc Castaignos pulled one back for Twente in the 88th minute, they could not complete the comeback as Groningen held on for a triumph that sees them move into seventh.
Elsewhere, Zwolle are back up to sixth in the table following a 5-0 away win at Excelsior, which featured doubles from forward Stef Nijland and midfielder Mustafa Saymak.
