The pair had started the day at Brazil's Interlagos circuit separated by a single point, with Norris on pole position and closest rival Piastri third on the grid.
The Australian's hopes disappeared on lap six of 24 when he lost control on the wet kerb at turn three and spun into the barriers, in another huge moment in a championship of fine margins.
The race was then halted on lap eight, resuming with a rolling start after barrier repairs and Norris leading the field with Mercedes's Kimi Antonelli and George Russell in his slipstream.
Antonelli threatened to get past Norris, closing the gap to less than a second on medium tyres to the Briton's softs, but had to settle for second when Sauber's home hero Gabriel Bortoleto crashed heavily at the start of the final lap, triggering double yellow flags.
The Brazilian, whose car went airborne and careered into the walls on both sides of the track in a scary crash, reported over the radio that he was unhurt. He was taken to the medical centre and released, then he ran down the pit lane to the Sauber garage.
Antonelli ultimately took the chequered flag 0.845 seconds behind Norris, with Russell third and Mercedes regaining second place overall from Ferrari in the standings.
Red Bull's reigning world champion Max Verstappen finished fourth and is now 39 points behind Norris in third place overall.
Ferrari's Charles Leclerc was fifth, after overtaking Aston Martin's Fernando Alonso, with teammate Lewis Hamilton seventh and the final point going to Alpine's Pierre Gasly.
"It was tough. It makes the win more rewarding when you have a race like this," said Norris.
"Not an easy race but it's always a difficult one in Brazil. Mercedes were quick. I expected us to be a bit better but we weren't. I don't know how much of that is down to tyres, or did Mercedes do a good job?"
Piastri's race was the latest of many subpar performances from him in recent rounds, but he remained hopeful as he looked ahead to the qualifying for the main race.
"I will try to put this behind," said Piastri, who lost the overall lead to Norris in Mexico last month. "There's a lot more points on offer tomorrow. So the better job I can do in qualifying to get a good starting point, the better it will be."
Team boss Andrea Stella admitted to mixed feelings, saying: "It's just an episode. Oscar has been fast this weekend. We look forward to regrouping, repairing the car. The big points are tomorrow."