Ducati's Dovizioso held off Repsol Honda's Marquez on an intense final lap to win by just 0.027 seconds for the first blow in the fight for the title.
Dovizioso gave no quarter as four-time MotoGP champion Marquez attempted a typically audacious last corner pass. As his pit crew watched through their fingers the Italian resisted the Spaniard's robust challenge, powering across the line to delirious celebrations in the Ducati garage.
Veteran factory Yamaha rider Valentino Rossi finished in an impressive third place under the dark desert sky, ahead of Briton Cal Crutchlow's LCR Honda.
Frenchman Johann Zarco led early on, having taken pole position with a swashbuckling qualifying lap that smashed the desert track's record -- the longest standing on the MotoGP calendar.
But on the dusty circuit, which tends to grind away at tires, he began to struggle for grip and eventually had to settle for a disappointing eighth place on his Tech 3 Yamaha.
Wheel to wheel
As Zarco slipped back, the curtain raiser picked up where last season had left off, with all eyes on Dovizioso and Marquez. Even the added spice of veteran nine-time world champion Rossi, 39, tracking the pair closely, fresh from inking a new two-year contract, was not enough to distract from an enthralling duel.
"Towards the end, after passing Zarco, I tried to pull out a gap on Marquez and Rossi but I didn't have any more grip and was unable to increase my lead," Dovizioso told reporters after the race.
"At the last curve I had to be very careful because I knew that Marquez would try and pass me, and that's how it went.
"He closed the door on me even more than [last season] in Austria and Japan, but I passed him on the inside of the curb and then took advantage of the power of my Desmosedici to stay ahead."
Marquez said: "What a race. I'm really happy with our second-place finish because this is one of the tracks on the calendar where we struggle most.
"I tried to pass Dovi in the last corner but he beat me. Honestly, I was expecting that, but I had to try the move anyway, to see if there was any way we could make it happen. In the end, Dovi had something more than me and he deserves this win."
The popular Dovizioso, with eight wins to his name, becomes the second most successful Ducati rider of all time in MotoGP, after Australian two-time world champion Casey Stoner.
But it was an unhappy return for his teammate Jorge Lorenzo. The Spaniard was sent sprawling into the gravel in a crash caused by technical issues with his bike and was forced to retire.
Next in the MotoGP calendar, the teams head to South America for the second race of the season, the Argentine MotoGP on April 8.