Trailing 5-2 in the seventh, the Mariners put Kyle Seager and Kendrys Morales aboard with one out against Jerry Blevins and Raul Ibanez followed by cracking a three-run homer into the first row of the right-field stands.
Chavez moved up on second on fielders' indifference and Franklin (3-for-5, three RBI) then delivered a line drive into left-center, scoring both runners.
Franklin Gutierrez, playing in his first game since April 22 due to hamstring injury, belted a solo homer as Seattle evened its three-game series.
Yoervis Medina (2-2) earned the decision with 2 1/3 innings of shutout relief.
Jed Lowrie went 3-for-3 with three RBI for the Athletics, who fell into a first-place tie with Texas atop the American League West with their fourth loss in six games.