Dutchman Verstappen converted his pole position into a lead at the first corner and was able to keep Norris at bay for the full 19 laps.
“It was not too bad,” Verstappen said. “It feels a bit like old times. The whole race, Ferrari was always very quick, but for us we were finally racing again. We had good pace.”
Norris jumped up from fourth on the grid to second on the first lap with a superb first four corners, sweeping past Mercedes’ George Russell into the fast Esses after passing Ferrari’s Charles Leclerc at Turn One.
Sainz and Leclerc staged a race-long battle in the Ferraris in a sprint that was engrossing throughout.
The Ferraris staged a frantic scrap over the first five laps in which they seemed to spend almost as much time side by side as in line astern. Sainz time and again attemped moves only to be rebuffed, often as he ran into corners too deep.
Once ahead on lap five, with a dive down the inside at Turn 15, the Spaniard attacked Russell as the Mercedes driver started struggling with front tyre wear, and moved ahead at Turn 15 on lap nine, Leclerc following him through at the same place next time around.
Leclerc came back at Sainz in the closing laps as the Spaniard himself tried to attack Norris for second.
And Norris’ error into Turn One, locking his inside front wheel, allowed Sainz to slip by and reduced the pressure from Leclerc, who had been lining up for a move on his team-mate.
Norris said: “It was a good race. I am pretty happy with how things ended up.
“It was a tough one, I thought I could maybe hang on to second but Carlos did a good job.
“My front tyres were completely finished and there was not a lot I could do. Whether I’d stayed straight or defended, I am pretty sure I would have lost the place.
“I did the most I could, I don’t think we had the pace on Max and the Ferraris so I am happy to finish third.”
Russell faded to fifth place, but managed to hold off team-mate Lewis Hamilton on the final lap.
Haas drivers Kevin Magnussen and Nico Hulkenberg took the final points in seventh and eighth places.