The 36-year-old Sweeney took batting practice before Friday night's game against Cincinnati to test his sore lower back, and was available in the series opener.
The five-time All-Star began the day with a .258 batting average, six homers and 16 RBIs.
The 24-year-old French was 0-1 with a 7.45 ERA in three games for Seattle. He made one start after the Mariners recalled him from Tacoma on June 6 to replace injured right-hander Doug Fister.
Manager Don Wakamatsu says Fister is on track to return to the rotation next weekend at Milwaukee.