The world number 11, starting his round at the tenth, was four under par at the turn and followed that up with three birdies in the opening four holes of his back nine.
The seven-under, bogey-free round leaves the 28-year-old Northern Irishman two shots behind clubhouse leader David Horsey, 32, of England.
Needing an eagle at the par-four ninth to make a 59 round or a birdie to break the course record, Donaldson found the rough with his approach and then missed his seven-foot putt to save par.
Rory McIlroy said Wednesday that he will take a new approach to winning the Masters for the first time by slashing his Augusta support team which has ballooned to nearly 20 people in recent years.
Paul Lawrie made a promising start in the $3 million Omega Dubai Desert Classic - after only getting into the star-studded field an hour before teeing off.
McIlroy called an early end to a winless and injury-plagued season last October, but returned with a tie for third in Abu Dhabi last week and carried on where he left off on Thursday. McIlroy has averaged 67 in his five rounds this season and finished 18-under last week in Abu Dhabi.
"It's obviously great to be back here in Dubai defending our title", said Garcia, who started his 2018 campaign with a five-shot win in the Singapore Open last week.
"I knew Peter Hanson was struggling and, after hitting a few balls on the range this morning, he said he couldn't play", revealed Lawrie, who has his oldest son, Craig, caddying for him.
"I guess that has changed the way I view this week a little bit because of how I played last week".
"I am ahead of schedule right now".
"Trying to keep improving, keep getting better in all aspects of the game, just like I try to do every year".
"I don't think you could have got better conditions out there".
"It was the first European Tour event outside of Europe in 1989, and we have a wonderful list of previous champions".
"It's always one of the highlights of the year for me to come and play here with the memories that I have".
"I feel like I've got a bit of a spring back in in my step because of what happened last week and I'm very confident in myself".