With the game tied 4-4 in the sixth, Justin Smoak led off the inning with a single against Rays starter Chris Archer.
Wesley Wright came in to pitch and Ackley greeted him by belting a 3-2 pitch to the base of the center field fence for a triple, sending home the decisive run.
Brad Miller socked a pair of solo homers and Smoak finished 2-for-4 with a two-run single for the Mariners, who claimed the opener of a three-game series.
Erasmo Ramirez (4-0) got the win despite yielding four runs on seven hits over 5 1/3 innings.
Oliver Perez and Yoervis Medina combined for 1 2/3 scoreless relief innings and Danny Farquhar notched his fifth save to secure the victory.
Archer (6-5) was roughed up for five runs on nine hits in a five-inning performance for the struggling Rays, who came into the contest with a two-game lead over Baltimore for the final American League playoff spot.
Ben Zobrist smacked two homers -- a two-run shot and a solo blast -- for Tampa Bay.