Franmil Reyes, mired in a 1-for-27 slump entering the day, had a two-run single in the third when Cleveland scored four times to take control at 5-0. Andres Gimenez had two hits and an RBI, finishing 6 for 12 with six RBIs in the three-game series.
McKenzie (1-2) held the A's to three hits, struck out seven and walked one as Cleveland swept a series at the Oakland Coliseum for the first time since April 10-12, 2000.
''Really powerful stuff, really violent,'' Cleveland catcher Luke Maile said about McKenzie's outing. ''To call pitches for a guy like that is awfully fun because even when he's got something else in mind, you really can't go wrong when he executes well.''
It was quite a bounce-back weekend for the Guardians, who arrived in the Bay Area having lost seven in a row. They're 10-12 overall as they head home from a 10-game road trip.
''The game is fluid,'' manager Terry Francona said. ''We've been very streaky. To this point the down streaks have outweighed the good, but hopefully we're in the midst of changing that. We've played better baseball the last three days.''
The A's, who have lost six of their past eight, didn't advance a runner as far as third base until the seventh inning.
A three-run rally in the ninth helped Oakland avoid a shutout, as Christian Bethancourt and Kevin Smith delivered RBI doubles and Chad Pinder hit a sacrifice fly.
James Kaprielian (0-1), making his season debut for Oakland after being sidelined by a shoulder injury, walked four and didn't make it out of the third.
''It didn't go his way today,'' A's manager Mark Kotsay said. ''I haven't had a chance to talk to Kap yet, but it just looked like he was really laboring. And he didn't really have a feel for being able to command the fastball in the zone.''
Kaprielian was pulled after walking the first three batters in the third, and Reyes greeted reliever Jake Lemoine with a two-run single to right-center that made it a 3-0 game.
Reyes, who got a day off Saturday to clear his head amid his slump, had struck out in 14 of his previous 18 plate appearances before his run-scoring hit. He finished 1 for 5 with three strikeouts Sunday.
The A's optioned RHP Adam Oller to Triple-A Las Vegas to clear a spot for Kaprielian on the active roster. Oller began the season as Oakland's fifth starter but went 0-2 with an 11.17 ERA over three starts in his first taste of the majors.
Oller was acquired from the Mets as part of the deal that sent starter Chris Bassitt to New York.
Guardians: SS Amed Rosario missed a second consecutive game with a sore left wrist, though he hit in the batting cage Sunday morning. Francona expressed optimism that Rosario might return Tuesday against San Diego.
''We've got a day off tomorrow,'' Francona said. ''It will give him a chance to be OK.''
Athletics: C Stephen Vogt, on the 15-day I.L. with a sprained right knee, continues to rehab the injury and his timetable for return is unknown.
After a day off Monday, the Guardians will send RHP Zach Plesac (1-2, 3.80) to the mound Tuesday against San Diego to open a six-game homestand. RHP Daulton Jefferies (1-3, 3.26) starts for the A's on Monday in the opener of a three-game series against Tampa Bay.
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