Lawson actually finished in 13th place but was relegated to 17th because of time penalties incurred during the 57-lap race, one of which was for making contact with Lance Stroll's car.
George Russell edged out Lando Norris for second, but that placing is subject to a post-race investigation.
Piastri cut Norris' Formula 1 lead to three points after the fourth race of the season.
Russell was 15.599 seconds behind Piastri. Norris did well for third after a tumultuous race for him while Lewis Hamilton was fifth and Max Verstappen after having an early second pit stop.
Lawson's second five-second penalty was for causing a collision with the Sauber of Nicolas Hulkenberg, who finished 14th.
Russell's second placing is subject to a stewards' inquiry into whether the Mercedes driver used the drag reduction system outside the defined zone, Reuters reported.
"It's been an incredible weekend starting with qualifying yesterday and finishing the job today in style is nice," said Piastri.
He won from pole to finish, though his initial early lead was cut right back during a safety car period.
"It's very important given our owners. It's never been a track that's been kind to us, so it's nice to have our first win here."
- RNZ and Reuters
