The Europeans raced into a three-up lead after just four holes as Woods sprayed the ball all over the course, while Stricker's putting woes continued.
Stricker did card an eagle to grab a hole back at the fifth, but birdies from Donald at 8 and 9 saw Europe take a commanding four-up lead by the turn.
However, three birdies in four holes saw the Americans cut their deficit to just one by the time they had reached the 14th hole.
Donald played a wonderful wedge to within a couple of feet of the cup at 15, before tapping in to give them some breathing space.
Their comfort did not last long, however, as Woods hit right back at the next hole to leave America trailing by one at they entered the final two holes.
The world number two had started to find his range by now, stiffing his tee shot to give himself an excellent chance of a birdie at the par-three 17th.
Donald went one better with his approach, leaving himself a tap-in for a two. Both men knocked in their putts to set up a tense final hole.
Garcia and Stricker both had birdie opportunities on the final hole, but the American had the closer range.
The Spaniard rolled a 30-footer just to the right of the hole, giving Stricker the chance to earn a half in a game in which they had looked all but beaten.
Stricker could not hold his nerve to drop the birdie effort and Europe just edged over the line.