Albany tied with Team Lake Nona at seven-over par after the second round, before Woods and Poulter pipped Graeme McDowell and Henrick Stenson at the first play-off hole.
Woods fired a brilliant close-range approach, and Poulter – Europe’s hero at the last Ryder Cup – coolly rolled in the cup-winning put to clinch victory for his team.
Alongside Woods and Poulter, Albany's winning four-man team included Justin Rose and Tim Clark. Nona's unlucky losers also included Peter Hanson and Ross Fisher.
US Open champion Webb Simpson, meanwhile, won the individual event at the Isleworth Golf and Country Club with a round of 70.
It was a day of disappointments for McDowell, who was defeated by Simpson by a single stroke, while Woods carded a round of 73.
"I missed the cut last week [at Bay Hill] and I knew I wasn't that far off," Simpson said.
"I had some work to do and worked on it this weekend. I played great yesterday and played great today and I think it's going to give me confidence knowing that we're working on the right stuff. Hopefully that will carry into Augusta."
The event was extended into Tuesday following the delayed finish to last week's Arnold Palmer Invitational, despite being originally scheduled to finish on Monday.