"Welcome to being a 20-year-old," Red Bull team principal Christian Horner said. "Great start to a new decade for you."
His father, the former F1 driver Jos Verstappen, looked emotional as his son stood on the podium. Horner also appeared teary-eyed.
Dutch fans poured on to the grid to join the celebrations, with one fan applauding with a pair of clogs.
Vettel saved Ferrari's blushes with a superb drive but, as if Ferrari's luck could not get any worse, Williams driver Lance Stroll smashed into the side of him on the warm-down lap, cutting the Ferrari almost in half.
Verstappen won the Spanish GP on his Red Bull debut last year, and became the youngest driver on the front row when he qualified second for Monaco. He was still 18.
Until yesterday, however, his season had been blighted by engine problems. Since placing third at the Chinese GP, the second race of the year, he failed to finish seven times.
But his luck held this time, and he delivered a composed performance in keeping with the predictions that he will become a future great.