McIlroy, the U.S. Open champion who has had three top-five finishes in Abu Dhabi, had three birdies on his first four holes but erratic driving led to two bogeys on the next four. He steadied himself with three birdies on his back nine, including a chip-in on No. 8 from just off the green.
"It's a nice way to start the competitive season, I suppose," McIlroy said. "I didn't feel like I played that good. I definitely didn't strike the ball as good as I have been the last couple of weeks. I think it's just because your first competitive round of the season, card in your hand, you can get a little bit tentative or a little apprehensive."
Woods played a solid round of bogey-free golf that produced few momentous shots and two birdies on the National Course at the Abu Dhabi Golf Club, which was playing much tougher than in the past with narrower fairways and thicker rough. He missed several birdie chances, including a 6-footer on his ninth, the 18th hole. He also struggled with his approaches shots, resulting in many 25- and 30-footers that he couldn't sink.
Robert Karlsson (67) was tied for the lead, with Gareth Maybin and Richard Finch one shot back.
Sergio Garcia (71) had a hole-in-one on the 12th hole but struggled otherwise. Top-ranked Luke Donald, who played alongside Woods and McIlroy, also shot a 71.
McIlroy, who calls Woods a friend and was chatting with his playing partner for much of the day, admitted he didn't take any satisfaction from beating him in the first round.
"If it was the last day of the tournament and you're both going in there with a chance to win, I would take a lot of pride from that obviously," said the 22-year-old Northern Irishman, who has talked of idolizing Woods as a teenager and following him during a Dubai tournament when he played as an amateur in 2006 and 2007.
"But the first day of a tournament is a little different," he said. "You're just going out there and playing and seeing what you can do. But hopefully I can get myself into position where I do play with him on a Sunday and see how I get on."
Woods said he was satisfied to be challenging for the title going into the second round.
"Hit the ball well all day today. It was a good ball-striking round," Woods said. "I had a hard time reading the greens out there. The greens were pretty grainy and I just had a hard time getting a feel for it. Toward the end I hit some pretty good putts but overall I got fooled a lot on my reads."
Coming off a seven-week layoff, Woods has said he is fitter than he has been in years and brimming with confidence following his dramatic victory at the Chevron World Challenge last month. That ended a two-year run without a win. Prior to last month's win, Woods finished third at the Australian Open, and then delivered the clinching point for the American team in the Presidents Cup.
Since Chevron, Woods has moved up to 25th in the world after briefly falling outside the top 50 last year.
"It felt the same as it had from Oz to the World Challenge to here," Woods said of his game. "I controlled my ball all day and just had a hard time getting a feel for these greens. They are grainy enough where I just didn't quite read them right, and I hit them good, and then the grain would take it, not take it. It was just difficult."
Second-ranked Lee Westwood (72) and fourth-ranked Martin Kaymer (77) got off to poor starts and never challenged for the lead.
Westwood had four bogeys to go with two birdies on the front nine. Kaymer, who shot 24-under 264 last year to win the tournament for the third time, started with a double-bogey when his drive went out of bounds.