The Spaniard took the lead courtesy of a gorgeous tee shot into the second green, leaving him with three feet for his birdie.
He sunk that to go one up, but was hauled back to all square when he pushed a putt wide at the next hole.
The American took the lead with a birdie at the eighth, but an eagle three at the 10th restored parity once more.
Furyk got his nose back in front with a birdie at the 12th, but a par at the 13th cost him as Garcia squared the match.
The American led by one as they teed up at the par-three 17th. Furyk came up short of the green and pitched to 10 feet, leaving himself a tricky one.
Garcia found the green with his first shot, before lagging his putt next to the hole. That was enough to level as the players headed to the final hole.
Furyk looked tight as he found the rough at the back of the green, while Garcia finished on the front of the dancefloor.
Furyk's putt from the fringe was a tough one, with a downhill break, but he did well to get it within five feet.
The Spaniard missed his long one, meaning that Furyk had to hole for a half. The drama was not over, however, as Furyk pushed wide to surrender a valuable point to the Europeans.