The Liverpool striker struck after 78 and 81 minutes to give the Dutch a 14th consecutive victory in qualifying matches after Hungary had hit back from an early setback to threaten an upset in a thrilling Group E encounter.
They lead the group with a maximum 18 points, although the sight of Robin van Persie going off with a knee injury was a worry for the Dutch and his club Arsenal.
"Matches like this you won't see often," Dutch coach Bert van Marwijk, whose side thrashed Humgary 4-0 in Budapest at the weekend, told reporters.
"Attractive for the crowd with a lot goals but it was very tough as we had to come back from two setbacks.
"We started well but after we failed to add a second. We had the chances but our game became inaccurate and careless and that gave them the feeling something was possible tonight."
Van Persie opened the scoring after 13 minutes when two Hungarian defenders failed to clear a corner kick and the striker netted at the near post.
However, he was forced off a minute before half-time when he was injured in a tackle by Vilmos Vanczak.
Hungary took advantage of a sloppy start to the second half by the hosts to turn the match on its head.
First Gergely Rudolf dribbled into the area and his shot deflected into the net off Rafael van der Vaart and four minutes later a fine cross from Balazs Dzsudzsak was volleyed home by Zoltan Gera.
Playmaker Wesley Sneijder equalised on the hour after a lucky one-two with Vanczak and 12 minutes later Ruud van Nistelrooy put the Dutch ahead with a deflected shot after a superb solo run by Ibrahim Afellay.
Gera equalised with 15 minutes remaining after more Dutch uncertainty at the back but Kuyt volleyed his side back ahead from an Afellay cross and sealed the points with a flukey lob over the stranded Hungary keeper Marton Fulop.
Van Persie, who has suffered from knee problems in the past, was not too badly injured, according to Van Marwijk.
"It didn't look very serious but it was too painful for him to carry on as he was hit by a knee just above his own knee," the coach said.
After an encouraging start to their qualifying campaign Hungary are now in third spot with nine points from six games, level with Sweden who beat Moldova 2-1 and who have played two fewer games.
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