The American pair won their morning fourballs by trouncing Marc Leishman and China's Li Haotong and looked on track for another convincing win in the afternoon alternate shot format against the Australian and Mexico's Abraham Ancer.
But after being five up on the 10th, they went to sleep with the International pair storming back to halve the match on the 18th in an astonishing recovery.
It ended world number four Thomas' winning run at Royal Melbourne. He has now earned 3.5 points in his four matches, as has debutant Ancer.
After relying on Thomas over the first three sessions, his teammates finally stepped up to help the US win the foursomes 3-1 after losing the fourballs 2.5-1.5.
It leaves the biennial match play event finely poised, with Ernie Els' team 10-8 in front going into Sunday's 12 singles.
Fowler's approach shots were superb, allowing Thomas to finish off the good work with the putter on the front nine before Leishman and Ancer flipped a switch, winning five holes, including the last three, for an unexpected half point.
"Pretty amazing. Even when we were down on the front nine there, our spirits got a little low there for a few holes and the crowd was still getting behind us," said Leishman.
"We've spoken about it a lot but I think that's what got us through. I mean, walking to that 18th tee there felt like the Tour de France, everyone screaming, it was awesome."
Ancer added: "We kept believing."
Woods again opted not to play and focus on his captaincy duties as he tries to prevent the Internationals claiming only their second ever Presidents Cup after a lone victory in 1998.
Veteran Adam Scott, who was the last in after the morning fourballs, had barely half an hour to freshen up, grab a sandwich and go again for another 18 holes in the opening foursomes group.
But the American pair slowly started finding their range and by the seventh were in front with the tense match culminating in a 2-up win for the US partners.
Patrick Cantlay and Xander Schauffele, who were convincingly beaten in the morning fourballs, also secured a point.
They bravely battled back after going three down to Australia's Cameron Smith and South Korean Im Sung-jae for a gritty 2 and 1 victory.
The final match between Matt Kuchar and Tony Finau and An Byeong-hun and Joaquin Niemann went to the wire.
An had a putt to win it on the last but narrowly missed with the matched halved.