Salazar gave up two runs and seven hits. He is expected to be Cleveland's No. 3 starter.
Joba Chamberlain pitched a hitless ninth for a save.
Khris Davis tripled and doubled for the A's and finished with three hits. Davis, acquired by Oakland from the Milwaukee Brewers in a trade on Feb. 12, has been heating recently.
"He is swinging the bat well at the right time, as we get closer to the season," manager Bob Melvin said of Davis, who raised his spring average to .304.
Yonder Alonso hit his first home run and singled for Oakland.
Indians: Manager Terry Francona was pleased with Salazar's effort.
"We got him through six (innings) and we were managing his pitch count," Francona said. "He made a couple of mistakes, but he is starting to pitch through the fatigue."
Hahn, making his fifth start, has struggled in his last two outings, including Monday, when he gave up all six runs on 10 hits. He is attempting to bounce back after suffering a strained right forearm in mid-July that kept him out the rest of the season.
MONEY IN THE BANK:
Naquin cried tears of joy late last week when Francona informed him that he had made the opening day roster.
Naquin rebounded from strikeouts in his first two at-bats on Monday to connect for an opposite-field homer to left.
"You can see his confidence rising," Francona said. "If he continues to play like that, gain that experience, it's like putting money in the bank."
The Indians came up with one of the better defensive plays of the spring in the fifth inning.
After Alonso led off the inning with a single, Marcus Semien hit a sharp grounder that appeared destined for center field. Second baseman Michael Martinez backhanded the ball and flipped it to shortstop Jose Ramirez in the same motion. Ramirez caught the ball and fired to first for the double play.
Indians: OF Michael Brantley (right shoulder) will suspend hitting activities for the next few days and likely will be out for the remainder of the spring. ... OF Lonnie Chisenhall (right forearm tightness) returned to action on Sunday but is expected to be placed on the 15-day disabled list to start the regular season, retroactive to Monday.
Athletics: Melvin wants to make sure LHP Felix Doubront stays loose and flexible as he makes a bid for a spot in the starting rotation, so Doubront will start in a minor league game on Tuesday. "We have to keep him stretched out," Melvin said.
Athletics: RHP Sonny Gray, who will make his third consecutive opening day start against the Chicago White Sox on April 4, will oppose 2015 NL Cy Young Award winner Jake Arrieta and the Chicago Cubs on Tuesday at Sloan Park in Mesa.