American Jordan Wilimovsky, world gold medallist in Kazan in 2015 but only fifth in Rio, took silver and France's Marc-Antoine Olivier the bronze.
Olympic bronze medallist Olivier had won 5km gold on Saturday at the venue, some 130km south of Budapest.
Dutchman Weertman touched out after one hour 51 minutes and 58.5 seconds in the water, making the most of his longer reach to beat Wilimovsky by a mere tenth of a second in a three-man dash for the finish.
The gold was a first for the Netherlands in the event since it first featured at a world championships in 2001.
Britain's Jack Burnell, disqualified in Rio, finished fourth and 1.5 seconds behind Olivier after being second at the halfway stage. The top six swimmers all finished within four seconds of each other.
The women's 5km event is on Wednesday with the main swimming programme in the Budapest pool starting on Sunday.
(Reporting by Alan Baldwin in London, editing by Neville Dalton)
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