Ferrari's Charles Leclerc came in third. The victory was Piastri's third for the season.
With the win in Jeddah, Piastri took over the lead in the World Championship total, ahead of teammate Lando Norris, who had to settle for fourth place after a difficult starting position following Saturday's crash in qualifying.
Piastri stands on 99 points, ten more than Norris, who has a two-point lead over Verstappen.
Piastri also won in Bahrain last time. Before this season, he had only won two races in his career. He is the first driver from Australia to lead the World Championship series since Mark Webber 15 years ago.
Webber is now Piastri's agent.
Very well done by everyone. It got a bit more difficult in the start than I had thought, but it turned out very well. The car was superb, said Piastri over the team's communication radio.
McLaren has won four races so far this year. The only one who has managed to break into that dominance is Verstappen, who won in Japan.
I gave it my all, but that's how it is. We'll keep working, says the Dutchman.
Verstappen started in pole position, but was given a five-second time penalty for an infringement that affected Piastri at the beginning of the race.
The event was the fifth of 24 for the season. The next race will be decided in Miami, Florida on May 4.
