The world number 125 sunk birdies in four of his five opening holes, eagled his eighth and picked up a further shot a hole later to go into the turn at seven-under, vying for the lead with twice major winner Rory McIlroy.
Quesne bogeyed his tenth - a par four - but recovered with two more birdies to move to eight-under before a double bogey at the last left him three shots adrift of McIlroy's 63.
"I really enjoyed this one, I hit a lot of very good shots, holed some putts and one bad shot on the last one, but that's golf - I'm not superman," Quesne told Reuters.
Quesne is France's No.4 golfer, behind Victor Dubuisson, Gregory Bourdy and Raphael Jacquelin, but the 33-year-old said he was more concerned with improving his overall standing than becoming than becoming his country's number one player.
"I'm looking towards the world rankings so I can play a major, which I've never done. It's my goal this year," said Quesne, a late bloomer who only took up golf aged 17 and turned professional six years later.
He remained on the challenger circuit for several years, but joined the senior tour in 2010 before an unhappy season led him to lose his card.
Quesne bounced back to become a fixture on the European tour from 2012, winning that year's Andalucía Open as well as the 2013 Italian Open.
He has made a mixed start to 2014, claiming joint-eighth at Durban's Volvo Golf Champions, but trailing in 52nd in Abu Dhabi and 28th in Qatar earlier this month. Dubai, the final leg of the so-called Gulf Swing, has proved more agreeable.
"The course is in very good condition, the greens are amazing. As the day progressed the greens got faster so we have to manage this," said Quesne.
When asked whether he expected to be high up the leaderboard after Friday's second round, Quesne was phlegmatic.
"I just try to play my game and we will see," he added.