Kazmir gave up 13 hits and five runs over five innings for the Cleveland Indians in a 10-5 win against the Seattle Mariners on Saturday.
Beavan was tagged for nine runs and 16 hits in 5 1-3 innings.
Kazmir, trying a comeback on a minor league deal, had not allowed a run in three previous spring outings. He struggled in the first, giving up two runs on four hits before getting an out.
The two-time All-Star escaped jams in his next three innings but couldn't finish the fifth without damage.
"An outing could make the difference," Kazmir said. "I knew what to expect. I knew about it (the competition) coming in, so it's nothing different from now. Just continue to progress and get better."
"This outing right here, I wish the scoreboard looked a little better. But at the same time, this is spring training. I've done this before," he said.
A meeting between manager Terry Francona and Indians general manager Chris Antonetti on Sunday could decide whether Kazmir wins the job, or it goes to right-hander Carlos Carrasco.
Beavan is among four pitchers competing for two spots in Seattle's rotation.
"I was just having trouble getting the ball down. Whether it was a good or bad pitch, when I got ahead I just couldn't put them away, and that's usually not a problem," Beavan said. "It's just one of those you want to look past and good thing you got it over with in spring training. That was a fiesta today. I couldn't believe it myself."
Jason Giambi and Cord Phelps homered for Cleveland. Raul Ibanez homered and drove in three runs for the Mariners.
Mariners catcher Jesus Montero left in the second inning after being hit with a bat on a follow-through swing by Lindor. Montero had a bump on the right side of his forehead but no concussion.
Montero was able to stand up after a couple of minutes but was taken off the field on a cart.
Manager Eric Wedge said Montero should be OK.
"We'll see how he comes in and go from there," Wedge said.
Cleveland right fielder Matt Carson ran into the wall chasing Kyle Seager's triple in the fifth. He had a cut on his forehead and a neck strain, and was carted off.
NOTES: Indians closer Chris Perez, shut down since leaving his Feb. 26 appearance with a right shoulder sprain, pitched one inning against Class A hitters. He allowed a run on two hits, walked one and struck out one, and is expected to pitch again Monday against Colorado ... The Mariners, at the request of both players, released RHP Jon Garland and 1B Mike Jacobs. Garland had been in the running for a spot in the starting rotation but exercised an out-clause in his contract with the club. ... Former Mariners star John Olerud will have his No. 18 retired by his alma mater, Washington State, on March 30 when the Cougars play Stanford.