Two goals in three second-half minutes from Siem De Jong and Daley Blind put the Amsterdammers in control against Alkmaar, but Markus Henriksen pulled one back two minutes after the break.
De Jong got his second just before the hour mark and, even though Jozy Altidore reduced the deficit to one again in the 74th minute, Ajax held on.
The win took them a point above PSV Eindhoven and kept them a point ahead of Feyenoord, who won 2-1 at home to FC Utrecht.
Graziano Pelle put the hosts in front after 32 minutes, only for Cedric van der Gun to level in the 59th minute.
But Darryl Janmaat restored Feyenoord's lead two minutes later to keep them firmly in the title hunt.
FC Twente won 3-0 at 10-man FC Groningen.
Leroy Fer opened the scoring on the stroke of half-time and Groningen's hopes took another blow when goalkeeper Marco Bizot was sent off in the 78th minute.
And the visitors took full advantage, with goals from Dusan Tadic (80) and Luc Castaignos (87) wrapping up the win.
Elsewhere, Willem II slipped closer to relegation as the bottom side were thumped 4-0 at NAC Breda.
Goals from Kees Luyckx, Mats Seuntjens, Elson Hooi and Danny Verbeek left the visitors five points adrift.