The largest previous lead entering singles in the current format was 11-5 by the Americans in 2021 and Europe in 2004.
Europe shook off over-the-top hecklers in the New York crowd to seize command, needing only two wins and a draw from Sunday's 12 concluding singles matches to retain the Cup.
The Americans must reach 14.5 points to reclaim the trophy.
Tensions rose in Saturday's four-balls (best ball) matches.
World number two Rory McIlroy, who cursed at hecklers as they tried to distract him while putting, joined Shane Lowry for a 2-up win over Americans Justin Thomas and Cam Young.
England's Justin Rose and Tommy Fleetwood beat top-ranked Scottie Scheffler and Bryson DeChambeau 3&2, making Scheffler the first US player to start a Ryder Cup 0-4.
Rose got into a spat with Bryson DeChambeau's caddie, Greg Bodine, on the 15th green, but Rose called it a misunderstanding that was settled.
US Open champion J.J. Spaun birdied the last two holes to combine with fellow American Xander Schauffele for a 1-up victory over Jon Rahm and Sepp Straka. It was Rahm's first Cup pairs loss since 2018.
Tyrrell Hatton, a late replacement for injured Viktor Hovland, birdied the last for a 1-up victory with Matt Fitzpatrick over Americans Sam Burns and Patrick Cantlay.