Marquez's win was the 300th premier class victory for the Honda team and came 53 years after Jim Redman, of then Rhodesia, claimed the first one at Hockenheim.
Dovizioso battled hard to finish second behind the Spaniard in his works Ducati, while team mate Danilo Petrucci joined him on the podium in third.
Marquez was pleased with his consistent performance after equalling team mate Jorge Lorenzo's premier class win tally of 47, the joint-fourth highest.
"In Le Mans it's always difficult especially with the temperature. I think it was my first race with the soft tyre in the front but it was the safer option and I was trying to be consistent on the laps," Marquez said.
"I'm happy because normally on this track Ducati and Yamaha are very fast but we were there. This is good news for us."
Marquez and Jack Miller were involved in an early battle and the Australian surged ahead on the fifth lap, but the champion regained the lead two laps later and pulled away from the chasing pack to secure the victory.
Rins, who was second in the standings behind Marquez before the race began, finished 10th after starting in 19th following a disappointing effort in Saturday's qualifying.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Toby Davis)
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