His sixth birdie of the day on the 14th hole gave Molinari a share of the lead, but a pair of bogeys on the final four holes left him two behind joint tournament leaders Adrian Otaegui and Jordan Smith.
England's Smith sank five birdies and an eagle in a blistering 66, which was matched by Spaniard Otaegui's bogey-free round.
Reigning British Open champion Molinari finished the day in a share of fifth place alongside English duo Ross Fisher and Matt Wallace, Belgian Thomas Pieters and China's in-form Li Haotong.
Tommy Fleetwood, who needs to win in Dubai to have any chance of pipping Molinari to the Harry Vardon Trophy, was a shot behind his Ryder Cup team mate in a group including former world number one Rory McIlroy.
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Molinari felt there was room for improvement for both him and his playing partner Fleetwood.
"It's only Thursday so there are so many things that can happen," he said. "Both of us did not hit the ball our best but we managed to scramble well and close out two decent rounds.
"I don't think there is any benefit to thinking about what-ifs. It's too early."
World number 10 Fleetwood, aiming to finish at the top of the European Tour's money list for the second consecutive year, rued missed opportunities with the putter after a sluggish afternoon.
"After the first eight holes I could have been a few better," the 27-year-old Englishman said.
"I played lovely and burnt the edges on a couple of putts, so after eight it was a case of just stay patient and pick some birdies as you go along."
Spain's Jon Rahm opened his title defence with a 67 to sit one shot off the pace alongside England's Danny Willett.
(Reporting by Hardik Vyas in Bengaluru; Editing by David Goodman and Toby Davis)
