French (4-4), nursing his one-run lead through most of his seven innings, gave up a single with one out in the seventh to Shelley Duncan to end his no-hit bid.
He threw 99 pitches in seven innings, walking three and striking out four.
Aardsma allowed a one-out single to Michael Brantley then Asdrubel Cabrera followed with a single. Left-fielder Michael Sauders made a sliding catch on a fly ball by Shin-Soo Choo, with the runners holding. The runners moved up on a wild pitch. Then Duncan, in a nine-pitch at-bat, struck out on a high 95-mph fastball.
Fausto Carmona (11-14) allowed four hits and one run, walking five and striking out six in his third complete game this season.
French, acquired in a deadline trade with Detroit last July, was making his 20th career start, his eighth with the Mariners. Through the first six innings, he faced just one batter over the minimum, throwing 81 pitches.
He walked Lou Marson in the third with two outs. He then threw to first as Marson was breaking for second, catching him in a rundown.
He issued another two-out walk to Choo in the fourth and Duncan then flied out to left.
Of his first 18 outs, 12 came on fly balls and only one on the ground.
In the seventh, he walked Choo with one out. Duncan stroked his clean single just to the left of shortstop Josh Wilson.
Manager Daren Brown took the opportunity to visit French to make sure his focus wasn't damaged by the failed no-hit bid.
It only took French one pitch to get out of it. Jayson Nix lined out to Saunders. Choo, believing the ball would drop, was caught off second for a double play.
The Mariners had a promise of a big rally in the first inning but scored just one run. It turned out to be the only run necessary.
Notes: LHP Ryan Rowland-Smith rejoined the Mariners after a rehab assignment in the minors. He had been on the DL since July 28 with lower back strain. In 22 games, 19 starts, Rowland-Smith was 1-10 with a 6.96 ERA. He will work out of the bullpen and may have a spot start. ... Indians RHP Justin Masterson will miss his start Sunday to deal with a family matter. RHP Jeanmar Gomez will take his spot.