Beavan, 24, gave up one run on three hits over four innings in a 5-0 loss to the White Sox, putting his Cactus League ERA at 2.35 over 15 1/3 innings.
He's in competition with Hector Noesi, Hisashi Iwakuma, Kevin Millwood and Erasmo Ramirez for the three final starting spots behind Felix Hernandez and Jason Vargas, and Beavan has said from the start this spring that he knows there are no guarantees.
"I feel like I'm doing everything I can, everything I can control, and giving myself the best opportunity to make the decision hard for those guys," Beavan said after Saturday's outing. "All anybody can do is just go out there and perform and compete and leave it up to those guys to make decisions at the end of the day, because they know what's best.
The 6-foot-7 right-hander gave up a double and run-scoring single in the first to the White Sox, but then settled in and retired 11 of the final 12 batters he faced, seven of those on fly balls.
He said he was overthrowing early, bouncing a couple curveballs into the dirt in the first inning, but became more effective when he backed things down at the advice of pitching Carl Willis.
"Carl said, 'Stop trying to throw it too hard and just get a couple over for strikes and then you'll get your feel,'" said Beavan. "Once I did that a couple times I got back in that rhythm of staying on top of the ball and not trying to choke it, and it ended up working good for me."
Manager Eric Wedge said Beavan has already been stretched out enough that his workload was limited to four innings, even though he'd thrown just 48 pitches.
"We wanted to pull him back," Wedge said. "With our trip to Japan and everything, we've already got him built up to 80 [pitches]. So he's in good shape. We wanted to see some of these other guys."
What did he see from Beavan?
"I thought he threw the ball well," said the skipper. "He was a little out of sorts there in the first inning, but he fixed it and threw the ball well. He did a good job with his fastball again today.
"He trusts his fastball, he pitches with confidence. His secondary stuff continues to get better. He's still learning how to pitch, but for a young man, he's further ahead than most."