He will also become only the second playing captain in the Presidents Cup, after Hale Irwin did so in the inaugural matches 25 years ago.
Since returning in early 2018 from a fourth surgery on his back, Woods has won three tournaments, including his 15th major when he won the Masters in April.
"It's going to be a lot of work, but (it is) something I've been looking forward to for a long time," Woods said.
In October, Woods equalled the PGA Tour record with his 82nd career victory at the Zozo Championship in Japan.
He said the trip to Japan - the travel as well as winning - convinced him he was fully healed from knee surgery he had in August and capable of playing in the Presidents Cup, which will be held from December 12-15 at Royal Melbourne in Australia.
Woods also selected Tony Finau, US Open champion Gary Woodland and Patrick Reed.
The rest of the US team is Dustin Johnson, Xander Schauffele, Bryson DeChambeau, Justin Thomas, Matt Kuchar, Patrick Cantlay and Webb Simpson.
That leaves room for one more player, which could be current world No 1 Brooks Koepka, who had stem cell treatment in his left knee after the Tour Championship.
In announcing himself as his fourth and final pick, Woods said: "As captain, I'm going to choose Tiger Woods as the last player on the team.
"He's made ... nine Cups and he's played in Australia twice in the Presidents Cup, so this will be his third appearance as a player.
"And I find it interesting talking in the third person."
