The 32 year old went one up at the first when Quiros hooked his drive and double bogeyed, lost the second with a bogey but birdied the third to restore his advantage.
Quiros levelled with a par at the seventh, and went ahead for the first time with a birdie at the short tenth.
But Garcia birdied the 13th and 17th, and with Quiros losing the 14th after a bogey it was The Ryder Cup star who made a winning start to the group stage.
“I guess you could say it was a good battle, because it was a battle,” said Garcia. “That's about it pretty much.
“I think the one that has the most credit today is the people that were following us for not walking off after nine probably, because I can't really imagine that that was a lot of fun to watch. We weren't playing very good.
“I had a stretch there on ten, 11, 12, where he did hit some good shots. I made a big putt on 11 to stay one down, and then he had a great chance on 12 from about probably five feet, no more than that, to go two up.
“If he would have made that, the way I was feeling, I would have struggled. You could see that he felt that, because after that hole, he struggled the next couple. He hit a couple bad drives and I managed to win both of them and got one up.
“I was fortunate enough to hit three really good iron shots coming in on 15, 16 and 17. So that was about it.”