Spaniard Sainz, who had started on pole, prevailed for a third place finish with Leclerc fourth and only 0.184 of a second behind.
Fred Vasseur, experiencing his first Monza as Ferrari principal, had told his drivers in the closing stages that they could race without taking risks but they may have had a different interpretation to the Frenchman.
For Ferrari fans the battle was a highlight of a race that ended in a Red Bull one-two and Max Verstappen taking a record 10th win in a row.
Leclerc pushed hard to get past, both cars locking up into turn one with smoking tyres, leaving Ferrari fans with hearts in mouth.
It's always been a pleasure to race Charles whenever we've had the chance," Sainz told reporters after a wave of red-shirted fans spilled out of the grandstands to celebrate in front of the podium.
Persons:
Max Verstappen, Claudia Greco, Ferrari, Carlos Sainz, Charles Leclerc, Spaniard Sainz, Leclerc, Fred Vasseur, Frenchman, It's, Charles, we've, Sainz, Vasseur, Alan Baldwin, Toby Davis
Organizations:
Prix, Autodromo Nazionale Monza, Prix REUTERS, Monza, Ferrari, Spaniard, Thomson
Locations:
Monza, Italy