In a tense and incident-filled contest, Norris won by 10.388 seconds ahead of Mercedes’ teenage Italian rookie Kimi Antonelli, with the four-time champion Verstappen just 0.362 seconds behind after a charging drive, including a puncture, to a podium finish.
George Russell came fourth in the second Mercedes ahead of the luckless Piastri and three remarkable rookies in Haas’s Oliver Bearman and the Racing Bulls of Liam Lawson and Isack Hadjar. Nico Hulkenberg was ninth for Sauber and Pierre Gasly 10th for Alpine.
It was Norris’s seventh win of the season and the 11th of his career, moving him within sight of a first world title with a comfortable advantage ahead of Piastri, who fought hard and collected a 10-second penalty, with three rounds remaining.
"I was pushing and it was an amazing race," said Norris, who dedicated his win to the late Gil de Feran, a former driver who nurtured his early career.
"It’s so nice to win in Brazil at this amazing track and amazing fans. This is for Gil…. My mentor a few years ago. A perfect weekend."
Asked about the title race, he said: "I'm not thinking about it, not yet, not at all. It’s a great win, but to see Max and how quick he was today. That's where my mind is at. And there is a long way to go."
Norris now has 390 points in the title race ahead of Piastri on 366 and Verstappen on 341.
Verstappen said: “The race was full of action and I didn’t expect to finish on the podium after starting from the pit-lane and having a puncture. An incredible result to be proud of. We never give up."
Antonelli, who resisted intense pressure, said: "It was very stressful in the last few laps with Max on fresh tyres but I stayed cool to take P2."
Norris made a clean start from his 15th career pole to lead Antonelli.
Behind him, Gabriel Bortoleto rammed his Sauber into the barriers after being clipped by Lance Stroll. A Safety Car was deployed while the luckless local hero was rescued unharmed, just a day after an airborne crash in Saturday’s sprint race.
Lewis Hamilton, who lost his front wing in the mayhem, pitted, but worse followed for Ferrari when, on the restart, Piastri attacked Antonelli for second and the Mercedes bounced into Leclerc’s Ferrari, causing the Monegasque to retire.
Further back, Verstappen was 13th, but he had a puncture and pitted for mediums. He rejoined 18th - and charged.
A brief virtual safety car (VSC) period followed before racing resumed as Piastri’s move was investigated. He was given a 10-second penalty.
Ferrari called Hamilton in on lap 15 as Verstappen rose to ninth by lap 18 and then seventh when others pitted. His pure pace, 24 hours after saying his car was undriveable, was astonishing. By lap 25, he was fourth,.
Norris pitted from the lead on lap 31 for softs, giving Piastri the lead, and rejoined behind Verstappen in fourth.
Making use of fresh rubber, Norris swept past Verstappen on lap 33 to regain third. That soon became second when Verstappen and Russell pitted, rejoining 12th and sixth.
After nine laps as leader, Piastri pitted for softs and served his penalty, emerging eighth as Ferrari retired Hamilton.
Norris pitted again for mediums on lap 51. He came out third as Verstappen inherited the lead with Piastri second before he too came in and emerged seventh as the leaders began an 18-lap sprint to the flag.
Verstappen then pitted on lap 55 to rejoin fourth on softs ahead of Piastri, and the Dutchman passed Russell with an excellent move to secure a podium.