The victory takes them level on 78 points with reigning champions and long-term rivals PSV Eindhoven, each having played 32 of 34 league fixtures but with Ajax leading them by a goal-difference advantage of six.
Visitors Ajax took the lead 12 minutes into the second half when forward Arkadiusz Milik scored from a penalty controversially awarded for handball.
Milik secured all three points with an excellent finish from Mitchell Dijks's cross in the 64th minute.
PEC Zwolle had earlier climbed above NEC Nijmegen and into seventh place with a 2-0 home defeat of their rivals, strengthening their own pursuit of European football at their expense of their visitors' ambitions.
Australia goalkeeper Brad Jones was beaten twice, with Sheraldo Becker giving Zwolle the lead in the 17th minute when impressively curling into the top left from midfield having been under little pressure, before Queensy Menig added a second goal 10 minutes later, sending Nijmegen down to eighth.
There was also a surprise 2-0 win for 17th-placed De Graafschap at FC Utrecht, which strengthened their thin hopes of avoiding relegation.
A goal in each half from the talented Vincent Vermeij, in the 38th and 59th minutes, secured the three vital points, leaving Utrecht in fifth place, but third-placed Feyenoord beyond their reach. The 17th-placed De Graafschap are four points from safety, and have two games to play.
Feyenoord edged closer to securing third place with a 2-2 draw at Heracles, who are sixth.
Equalising goals from Michiel Kramer and Eljero Elia, after first Robin Gosens and then Wouter Weghorst had given Heracles the lead, leave them on 57 points, significantly behind the top two, and seven points above fourth-placed AZ who still have three games to play. Like AZ, Heracles also have 50 points.
Excelsior remain at risk of relegation after their 2-0 defeat at 13th-placed Twente left them in 15th.
They were close to securing a vital point when Jerson Cabral, who had missed a first-half penalty, gave Twente the lead in the 85th minute. Zakaria El Azzouzi added the hosts' second in the 90th.
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