The switch-hitting Headley drove a 1-2 pitch to left-center with two outs in the sixth inning for the Padres' second hit off Kershaw. The two-time Cy Young Award winner came within 18 innings of the major league record set by Dodgers right-hander Orel Hershiser, who threw 59 straight shutout innings in 1988.
Kershaw's astounding run included his no-hitter June 18 against Colorado. He also limited the Rockies to two hits over eight innings in the rematch last Friday at Coors Field.
The previous run against the 26-year-old left-hander came on an RBI double by Arizona's Aaron Hill in the third inning of Kershaw's 4-3 win June 13 at Dodger Stadium. His next outing was the no-hitter.
During his scoreless streak, Kershaw allowed 17 hits, struck out 52 batters, walked six and stranded 23 runners - four at third base. He retired the leadoff hitter in 36 of those innings.
Hershiser pitched scoreless ball over the final six regular-season starts of his only Cy Young season. He broke the previous big league mark of 58 consecutive scoreless innings, set in 1968 by another former Dodger - Hall of Famer Don Drysdale.