Pulling a gap of almost three seconds over Crutchlow who himself passed Zarco was a result of Dovizioso and Marquez both producing their fastest laps of the race on the penultimate lap.
Despite his Yamaha works team having a tricky pre-season, Rossi can still hope to be among the leading pack, though Honda's defending champion Marc Marquez and Ducati's Andrea Dovizioso remain the men to beat.
Andrea waited and stayed hidden; then, five laps from the end, he pulled the pin, snatching the lead from Zarco and taking command.
Italian great Valentino Rossi will start his 23rd season of grand prix racing, at the age of 39, from eighth place for Yamaha.
"I lost these last-lap last-corner [battles] in the worst circuits for us, Red Bull Ring, Motegi, here", he said.
Dovizioso re-took the lead on the final bend of the 22-lap night race to record a notable victory on a track where he had finished as runner-up for the past three years.
Italian Danilo Petrucci was third on the grid, 0.005 slower than Marquez, as the best of the Ducati riders on a non-works Pramac team bike.
Dovizioso said the pressure was on him going into the race, admitting there was an "obligation" for him to win in Qatar.
Marquez added that he is not anxious about losing a late-race battle against Dovizioso for the third time as these duels have happened on the Spaniard's weaker tracks.
"He overtook Zarco I just was behind him". This is the best way to start the season.
Rossi said he was happy with a podium finish, adding he wanted to follow it up with more this year.
Rossi, who won one race in 2017 and finished fifth overall despite breaking his leg at the end of August, had a good race compared to his Spanish teammate Maverick Vinales. "But at least when I have the light bike with the new tyre, I feel so good".
Making his 200th start in the premier class, Dani Pedrosa on the second factory Honda took seventh ahead of early race leader Zarco, Andrea Iannone (Suzuki) and Jack Miller who completed the top 10 in his first race for Pramac Ducati.
While there was elation on one side of the Ducati garage, the same couldn't be said for the other as Jorge Lorenzo crashed out of the opening race after battling for a top ten result.