With three of the five sessions of play completed, Rahm and Garcia remained the only European pair to have won a match, the Spanish duo recovering from losing the first three holes in Saturday’s foursomes to beat Brooks Koepka and Daniel Berger.
Narrow defeats for Paul Casey and Tyrrell Hatton, Viktor Hovland and Bernd Wiesberger and Lee Westwood and Matt Fitzpatrick meant the United States won their third straight session 3-1 to extend their overall lead to 9-3.
That meant Steve Stricker’s side required just five-and-a-half more points from the 16 available to regain the trophy, but Padraig Harrington’s side at least showed signs of being up for the fight in the afternoon fourballs.
Rahm and Garcia quickly moved three up on Koepka and Jordan Spieth, with Hatton and Shane Lowry one up after eight holes against Tony Finau and Harris English.
In the final match of the day, Rory McIlroy – who had been left out of a session for the first time in his career – and Ian Poulter were all square against Dustin Johnson and Collin Morikawa after five.
Europe’s perilous position could have been even worse had Rahm and Garcia not turned the opening foursomes match around in brilliant style, Garcia chipping in for a birdie on the ninth and also holing from eight feet on the 12th to move ahead for the first time.
The high-quality contest was then almost marred by an incident involving Koepka on the 15th, where the four-time major winner sought relief from a bad lie in the rough, claiming there was an impediment in his follow-through.
After being turned down by one referee Koepka called for a second opinion but did not get the answer he wanted and angrily said to the officials: “If I break my wrist it’s on f****** both of you.”
Koepka, who injured himself hitting a tree root and withdrew from the Tour Championship earlier this month, eventually took on the shot and even found the green, but a stunning approach to the next from Garcia set up a conceded eagle and the match was over on the next.
The win was Garcia’s 24th in total, edging him ahead of Nick Faldo, and extended his record points tally to 27.5.
Hovland and Wiesberger had been three up after six holes against Justin Thomas and Jordan Spieth before falling to a two-hole defeat, while Casey and Hatton had turned a four-hole deficit to just one when Casey holed out from 107 yards for an eagle on the 14th, only to bogey the next.
Westwood and Fitzpatrick’s 2 and 1 loss to Xander Schauffele and Patrick Cantlay meant Westwood has lost his last six Ryder Cup matches and Fitzpatrick is yet to win anything from his four across 2016 and this year.
Harrington had earlier defended his decision to leave former world number one McIlroy out of a session for the first time in his career.
McIlroy had played 26 straight matches since making his debut at Celtic Manor in 2010, but lost twice on the opening day – alongside Poulter and Lowry – without even reaching the 16th hole.
“Rory, we need him for the singles,” Harrington told Sky Sports. “Pushing him out this morning, making him play five times, would be an awful lot. Clearly you’re going to see him this afternoon.
“If we are going to win tomorrow we are going to need a big singles day. We have to keep that in mind.”