The Italian was two in front after the first three holes, but found himself pegged back by the time they reached the seventh tee.
A birdie at 10 saw Molinari retake the lead, but dropped shots at 12 and 13 allowed Woods to take the lead for the first time in the match.
Molinari struck back with a birdie at 14 to level the score, as a tense finish looked on the cards.
A bogey for the European at the penultimate hole saw him trail by one as they teed up at the last.
The players were forced to wait on the fairway for a while as Kaymer tidied up for a four to win the Cup.
Both men made the green, and when Woods missed his putt he conceded the hole and Europe were officially made the Ryder Cup champions of 2012 by virtue of an outright victory.