Rose got Team Europe off to the perfect start by birdieing the first but Johnson brought the match back level by sneaking in a birdie at two.
The third was halved before Kuchar birdied four holes in a row, winning three, to blast the Americans ahead.
Holes eight, nine, 10 and 11 were halved before Rose found a birdie three on 12 to bring the match back within two.
Kaymer had the chance to get himself a hole triumph after a close approach on the long par three 13th but the German couldn't find the putt.
Johnson then found a birdie at the short par four 15th after Rose slid his putt by from six feet to give America a dormie three lead.
Rose had to make his 20-foot putt on 16 but it trickled by to hand Kuchar and Johnson a 3&2 victory.
The point sees America take a 5-2 lead with one match left out on the course on the opening day.