Los Angeles swept the three-game series, winning its fifth in a row and its 10th in 11 games. With two days remaining in June, the Dodgers were six games below .500, but they have gone 31-7 since then to open a 7 1/2-game lead in the National League West.
Kershaw (11-7) did not give up a hit until the fifth inning and eventually allowed only three. He struck out eight and one of the two runs scored against him was unearned.
He also had one of the Dodgers' 14 hits, a two-out single in the second that brought home Andre Ethier and A.J. Ellis. Kershaw later scored on a double by Mark Ellis, who also contributed a two-run homer in the sixth.
Tampa Bay lost its fifth consecutive game, but the Rays stayed three back of Boston in the American League East.
Jeremy Hellickson (10-6) took the loss, giving up four runs on seven hits in just three innings.