De Jong opened the scoring after 20 minutes when he latched on to a precise long ball from Toby Alderweireld and Daley Blind added a second two minutes later when he linked up with Kolbeinn Sigthorsson before finishing at the near post.
Markus Henriksen pulled one back for Alkmaar early in the second half with a close-range header before De Jong restored the two-goal lead on 59 minutes when he curled a free kick round the wall.
US international Jozy Altidore netted with 16 minutes to play to set up a nervy finish, but the visitors held on.
Ajax top the table with 57 points after 27 matches, one ahead of PSV Eindhoven, who enjoyed a 2-0 win over RKC Waalwijk on Saturday.
“I am very satisfied with a win against an opponent we failed to beat the last two years,” Ajax coach Frank de Boer said.
“But it was totally unnecessary that it finished so close as we were in control with a two-goal lead but conceded their goals far too easily; they were gifts.”