The home side were held 1-1 when Kolbeinn Sithorsson took advantage of a mixup in the home defence to score for the champions after Luc Castaignos had given the home team a 21st minute lead at the end of a sweeping attack.
Twente moved to 19 points, one ahead of PSV Eindhoven who play on Sunday at Groningen.
Ajax, whose team had an average age of 22 years and 37 days, the youngest starting lineup in the league since September 2004, stayed third on goal difference with 18 points.
They are one point ahead of Feyenoord and Vitesse Arhem, who came from 2-1 down to beat Hereenveen 3-2 away.
Feyenoord could have been joint leaders had they not conceded a last minute equaliser at Go Ahead Eagles, for whom Marnix Kolder converted an 89th minute free kick to force a 2-2 draw.
Vitesse snatched a winner through left back Patrick van Aanholt, on loan from Chelsea, two minutes into stoppage time.
Utrecht moved away from the fringes of the relegation zone with a 4-2 win over NAC Breda while RKC Waalwijk remain second bottom after a 2-1 loss at home to Roda JC Kerkrade.