Alberto Callaspo hit a two-run homer and Yoenis Cespedes, Jed Lowrie and Brandon Moss added RBI hits for the Athletics, who broke out offensively after being shut out Monday in their big league record 10th straight opening loss.
But behind a strong performance from Kazmir (1-0), the A's followed an opening day defeat with a victory for the seventh time in those 10 years. Kazmir allowed just three hits and no walks in 7 1-3 scoreless innings, striking out five.
Corey Kluber (0-1) failed to make it out of the fourth inning and lost for the first time in 13 starts since June 27. He gave up five runs and eight hits in 3 1-3 innings.
The teams played the earliest doubleheader in baseball history following the first rainout in Oakland since May 5, 1998, on Tuesday night. The previous earliest doubleheader came on April 7 in 1971, '74 and '96, according to STATS.
Josh Linblom was called up from Triple-A Sacramento as Oakland's 26th man to start the second game against Zach McAllister. The Indians added reliever C.C. Lee to the roster for the second game.
The rain delayed Kazmir's Oakland debut by a day but it was well worth the wait for the A's as he looked like the pitcher who thrived down the stretch last season when he helped the Indians make the playoffs.
Kazmir parlayed that success into a $22 million, two-year contract as he replaced Bartolo Colon in the rotation.
Kazmir found his groove after allowing a pair of fly balls to the warning track in the first inning. He allowed only singles to Michael Brantley in the second and Jason Kipnis in the fourth, and a double to Mike Aviles with one out in the eighth.
Dan Otero gave up a two-out RBI single in the ninth to Carlos Santana that ended the shutout.
Oakland scored twice in the second inning to take a 3-0 lead despite losing a replay challenge. The disputed play came with runners on first and third and one out when Josh Donaldson hit a grounder to third. Carlos Santana threw home and umpire Mark Wegner ruled that Gomes tagged Derek Norris before he touched home.
A's manager Bob Melvin challenged the call but the ruling stood after a review of about 4 minutes, 45 seconds, because there was no conclusive evidence to overturn it.
Kluber got to throw some warmup pitches after the delay but allowed an RBI single to Lowrie to fall behind 3-0.
NOTES: Kazmir's outing follows Sonny Gray's six shutout innings in the opener, marking the first time since at least 1914 that the A's have started the season with two straight scoreless starts. ... The Indians have six sweeps and four splits in their previous 10 doubleheaders.