The victory was the Dutch driver's third of the season, and second in a row, and he made it happen with a crucial overtake around the outside on Sainz at the start to seize second place before blasting past Leclerc on the ninth of the 57 laps.
The Italian football team was named Team of the Year for the second time after winning the 2020 European Championship, having previously received the honour after lifting the FIFA World Cup in 2006.
Verstappen has won at Imola for the second year in a row.
The Dutchman crossed the line just half-a-second ahead of Leclerc, who keeps the championship lead.
Media reports have valued the new deal at 40-50 million euros ($44-$55 million) a year.
Lewis Hamilton, beaten this year on the last lap of the last race, did not attend the event at the Louvre museum in Paris.
Red Bull's Verstappen seized the title after overtaking seven-time world champion Hamilton on the last lap of the race amid uproar over how a late safety car period was conducted.
Red Bull rival Max Verstappen ended the Briton's quest for an unprecedented eighth F1 title with a safety-car aided, last-lap pass for championship glory at the season-ending Abu Dhabi GP.
Verstappen passed title rival Lewis Hamilton on the last lap. After a record 22 races, Verstappen had 395.5 points to Hamilton's 387.5.
The final race is in Abu Dhabi next weekend.