Laffey (2-2, 1.35 ERA) allowed only one unearned run on five hits, a walk and his own throwing error. The left-hander struck out six.
Jensen Lewis (2.82 ERA) gave up one hit over one scoreless inning before Betancourt (7.31 ERA) came on, seeking his fifth save. But the right-hander, who was so dominant in a setup role a year ago, struggled again. He got a quick out, then loaded the bases on a walk and two hits and was replaced by Kobayashi.
Kobayashi (1.86 ERA) uncorked a wild pitch allowing a run to score and putting A's runners on second and third.
The right-hander then struck out pinch-hitters Mike Sweeney and Ryan Sweeney on devastating split-finger fastballs for his second save.
While Betancourt wore a look of exasperation in the Tribe dugout after being lifted by manager Eric Wedge, Kobayashi never seemed the least bit flustered -- even after his wild pitch.
That was the most tell-tale sign of possible things to come. The 33-year-old's performance showed why the Indians signed him to a two-year contract last November -- and why he was able to record 227 saves in his native Japan over the past eight seasons.
Betancourt gave up three runs in the 10th inning Tuesday against Toronto and took the Indians' only loss during a seven-game homestand.
The unearned run on Laffey's first-inning throwing error broke 44 1/3 innings shutout innings by Cleveland's starting rotation.
It was the longest streak by Cleveland's starting staff in nearly 60 -- singe Hall of Famer Bob Lemon, Gene Bearden, Sam Zoldak and Hall of Famer Satchel Paige worked 47 straight scoreless innings in August 1948 on the way to the Indians' last World Series championship.
The longest streak in AL history was 54 innings by the Baltimore Orioles from Sept. 1-7, 1974. It ended at old Cleveland Stadium on a two-run single by Tommy McCraw against lefty Ross Grimsley.
This past week, Tribe starters went 6-0 with an 0.16 ERA. They haven't allowed an earned run in 50 1/3 innings, since C.C. Sabathia gave up a fifth-inning run to Toronto last Friday in a 6-1 win.
Cleveland got just enough offense to win for the eighth time in its last 10 games.
Ryan Garko (.234) had an RBI double in the first inning. Ben Francisco (.344) and Travis Hafner (.222) had RBI singles in the third and Jhonny Peralta (.221) led off the seventh with his sixth homer. Peralta scored three times and Francisco went 3-for-4, though one of his hits was an infield popup that just fell between Oakland pitcher Chad Gaudin and first baseman Daric Barton, who lost the ball in the sun.