Rose and Donaldson led the way with five-under 67s as winds gusting up to 20mph made scoring difficult in the opening event of the European Tour's three-week desert swing that also includes the Qatar Masters and Dubai Classic.
Sharing third place on 68 were Spain's Pablo Larrazabal and Thorbjorn Olesen of Denmark.
World number one McIlroy made the worst possible start when he somehow contrived to putt the ball into a greenside bunker at the 10th hole, his first. He escaped with a par but two double-bogeys later in the round meant he slumped to a three-over 75.
The 23-year-old Northern Irishman signed a massive sponsorship deal in Abu Dhabi on Monday but he refused to attribute his lacklustre display to his new Nike clubs.
"Of course, it takes time to get used to ... it's a little bit of an experimental period," the twice major winner told reporters. "I guess when you're going out with new stuff you're always going to be a little anxious.
"I just felt like I wasn't swinging that well. For the most part I was very happy with everything."
McIlroy's playing partner Woods made a trademark charge with three birdies in four holes from the 15th but then completely duffed his drive at the first, the American striking the ground behind the tee peg before squirting the ball out just 120 yards.
"My game plan was to hit a three-iron or five-wood on that hole and then I changed my mind," said the world number two after carding a level-par 72.
"I should have just backed off and followed my game plan. I had a strategy for the day and clubs I was going to use and what spot I was going to hit it to. I didn't do that there, paid the price and made bogey."
World number five Rose also made a hash of the first hole as he opened with a bogey five.
The Englishman recovered in style at the Abu Dhabi Golf Club, picking up birdies at the second, third, fifth, eighth, ninth and 10th.
"My short game was probably what separated me today," said Rose. "I holed a very long putt on the fifth, probably 50 or 60 feet, so that was clearly a bonus.
"Then I holed a bunker shot at the ninth. It was a very tough shot, one of those you up-and-down only about 50 percent of the time, so that was an extra bonus."
Rose said his game from tee to green was nowhere near its best.
"I didn't have it completely under control with the long game," he explained. "I didn't feel perfect.
"I think it was the type of day where you had to accept some mistakes. The course this week is set up very difficult, the fairways are narrow and, with a lot of crosswinds, the fairways are hard to hit."
Rose said he had always liked grinding out a score in tough conditions.
"That's the type of golf I like to play," he added. "I tend to like tournaments where the winning score is between eight and 12 under par.
"Those are the tests that I feel suit me well."
Donaldson followed up his tied 16th finish at last week's Volvo Golf Champions event in South Africa with a polished six-birdie display in the Middle East.
"This is a tough course, it's tricky out there, and I'm chuffed with the score," said the in-form Welshman. "I made a few good up-and-downs and holed a bunker shot.
"Five-under is a great start because if you're slightly off here you can be severely penalised. The rough is quite penal and you've got to keep it out of there."
Paul McGinley, appointed Europe's 2014 Ryder Cup captain on Tuesday, came back down to earth with a bang as the Irishman slid to a 76.
Also on four-over were defending champion Robert Rock of Britain and former Ryder Cup skippers Jose Maria Olazabal and Colin Montgomerie. Darren Clarke, the 2011 Open winner, recorded a 79.
- Sports & Recreation