Champions McLaren discovered an electronics issue with Norris's car before it was due to head out on track.
Piastri's car was then pushed off the grid and back to the garage.
Mechanics removed the floor on Norris's car, with the team still working to resolve the problem as the grid closed for the second race of the season.
The Briton had qualified sixth, with Piastri fifth but both grid positions were left empty.
"We also identified an issue on Oscar's car, which means we were unable to start the car from the grid," said a team spokesman.
"We have now returned the car to the garage to further investigate the issue there."
Piastri won in China last year from pole position.
But the 24-year-old has made a horror start to 2026, crashing his McLaren on the way to the grid at the Australian GP in Melbourne last weekend.
Mercedes locked out the front row in Shanghai on Sunday, with 19-year-old Italian Kimi Antonelli the youngest ever grand prix polesitter ahead of championship-leading teammate George Russell.
F1 is racing hours after it announced next month's races in Bahrain and Saudi Arabia will not go ahead because of the war in the Middle East.
