Italian great Valentino Rossi, who ended last season feuding with the young Spaniard, was a close fourth a day after announcing a contract extension with Yamaha.
Spaniard Lorenzo, whose future at Yamaha remains less certain, started on pole but the Ducati pairing of Andrea Iannone and Dovizioso then blasted past while Marquez dropped back.
Iannone led but then crashed out in a blaze of sparks after five laps and Lorenzo took over at the front four laps later and was not seriously challenged thereafter, with Dovizioso and Marquez trading places and fighting right to the last corner.
The race was the first of a new era, with Michelin replacing Bridgestone as tyre supplier and the introduction of standard electronics.
Lorenzo kissed the rear tyre of his bike after parking up at the finish, having made the right choice.
"To choose the right rear tyre was very important," he said. "I tried the warm-up with the hard one but I didn't get the best feeling. It was very risky to go for the soft one because our main rivals Marquez and Rossi chose the hard one.
"So finally I said 'Okay, the warm-up was not good so let's try with the other tyre' and finally it was good because in the last laps I could make the fast lap...so very happy with this victory," he told BT Sport.
Marquez, who had been well off the Yamaha's pace in testing, was also happy.
"I did my maximum," he said. "The perfect race was second place but it was really difficult to overtake because I lose a lot here in the acceleration on the straight...16 points for the championship is also important."
The next race is in Argentina on April 3.
(Writing by Alan Baldwin in London, editing by Brian Homewood)
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