The foursome will join the eight players who have already booked automatic places to play in the competition at Le Golf National in Paris, beginning Sept. 28.
Among the automatic qualifiers is Rory McIlroy who makes his fifth consecutive Team Europe appearance and whose participation this year will tie him with Darren Clarke for the most Team Europe appearances by a player from Northern Ireland.
Woods, who has five top-10 finishes this year, including a runner-up at the PGA Championship three weeks ago, ended up 11th in the final Ryder Cup standings.
Mickelson and DeChambeau also received at-large picks Tuesday and will join qualifiers Brooks Koepka, Dustin Johnson, Patrick Reed, Justin Thomas, Bubba Watson, Jordan Spieth, Rickie Fowler and Webb Simpson.
Furyk will add a final pick Monday following the BMW Championship at Aronimink Golf Club.