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.
U.S 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 told reporters.
"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."
Feyenoord beat Utrecht 2-1 on Sunday to move back to third on 56 points, above Vitesse Arnhem, who won 4-0 at ADO Den Haag a day earlier.
Italian Graziano Pelle put Feyenoord ahead after 32 minutes when he controlled a pass from Ruben Schaken with his right foot and finished with his left at the near post.
Utrecht drew level on 59 minutes following a blunder from Stefan de Vrij, who failed to clear and allowed Cedric van der Gun to score.
Darryl Janmaat, however, restored the lead two minutes later following a defence-splitting combination with Schaken.
"I thanked Darryl for his goal after my mistake gave them a goal," said De Vrij. "I knew he [Van der Gun] was behind me but I expected him on the other side, but mistakes like this should not happen."
Elsewhere, NAC Breda cruised to a 4-0 home win over neighbours Willem II Tilburg, who remain 18th with 17 points.
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