Kazmir, scratched from his previous scheduled outing with triceps tightness, gave up two runs and four hits in the first inning but only one run and no hits the rest of the way. He struck out five and walked three in 4 1/3 innings.
Vogt, a catcher who was serving as the designated hitter, connected in the third and fifth off right-hander Scott Baker, bidding for a spot in the rotation.
Logan Morrison drove in three runs for Seattle. He hit a run-scoring single in the first inning and a two-run homer in the fifth.
Stefan Romero added three hits for the Mariners.
Mariners: Baker missed a large chunk of the last two seasons after having Tommy John surgery on April 17, 2012. He made rehab appearances with Chicago Cubs minor league affiliates Kane County and Dayton in 2013 before starting three late-season games for the Cubs and compiling a 3.60 ERA.
Baker was inconsistent on Saturday, giving up six runs and seven hits with six walks and no strikeouts in four innings.
Athletics: Kazmir, who signed a $22 million, two-year contract as a free agent during the offseason, acknowledged he was erratic early against the Mariners. He settled in to throw about 85 pitches in his third Cactus League start.
"It feels good to get stretched out," Kazmir said.
The left-hander went 10 days without pitching in a game. He did throw a lengthy bullpen session.
"I tried to stay sharp," he said.
Now, the longer he pitches, the more he studies his mechanics.
"I didn't know much about mechanics three years ago," Kazmir said. "I just did it. I didn't know how I did it."
Mariners: Two of the club's highly regarded pitching prospects threw in a minor league game for Triple-A Tacoma against the Padres' Triple-A affiliate.
Right-hander Taijuan Walker, who has been experiencing shoulder soreness for most of camp, allowed a run and two hits in one inning. He threw 15 pitches, 10 for strikes, and hopes to join the rotation by late April.
Right-handed reliever Stephen Pryor, coming back from a torn back muscle, threw a perfect inning with one strikeout. He threw 10 of 11 pitches for strikes, but his status remains uncertain.
Athletics: Right fielder Josh Reddick, who left Friday night's game against the Giants in the early innings with a contact-lens issue, went 0 for 3 with a walk.
Oakland left fielder Yoenis Cespedes did not play. He is in a 2-for-30 slump (.067) over his last 10 games.
Left-hander Joe Savery appears to be a good fit for the A's bullpen after being claimed off waivers from Philadelphia on Feb. 17.
He made his seventh appearance of the spring Saturday, picking up his third hold. He gave up two hits and walked one. Savery is 1-0 and has given up only two unearned runs.