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After leading out Europe in both sessions and winning their fourball and foursome matches yesterday, the Anglo-Swedish duo again were chosen to go out first by captain Paul McGinley.
They didn't get off to the best of starts, though, as Kuchar rolled home his birdie putt after Rose and Stenson had both slid theirs by.
Rose made birdie at the par-five second to respond immediately before Watson notched his first sub-par score to restore the US advantage at the third.
At the sixth Kuchar found the cup from distance to extend the Americans' advantage to two holes, but that would be their last hole victory as Rose and Stenson between them incredibly birdied 10 holes in a row.
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Kuchar and Watson birdied five of those 10, but they couldn't keep up with the on-fire European duo, with Rose in particular making several important putts.
At the 10th he made a putt from off the green to halve before again equalling the American birdie at 11.
Stenson's birdie at the 12th then put Europe ahead in the match for the first time before Rose again found his range with the short stick at 13.
It looked as if Europe would go 3 up with three to play at the 15th when Stenson stiffed an approach to gimme range, but Kuchar made a huge putt from distance to keep the Europeans within two.
At the 16th, both Rose and Stenson were wayward off the tee, but Rose delivered a third shot to just a couple of feet that was good enough for the hole and a 3&2 victory. There was a Ryder Cup-record 21 birdies in total in the match.
Rose should play again this afternoon and will attempt to become the first European to ever win all five of his matches at a Ryder Cup. Stenson, however, is sitting out of the foursomes because of a tight back.