The Mariners had grabbed a 3-2 lead in the eighth on a Dustin Ackley RBI single, but the Rangers came right back in their half of the inning to tie the score on a run-scoring double by Elvis Andrus.
After Brad Miller fouled out, the Rangers intentionally walked Nick Franklin to get to Seager, but he crossed up the strategy by lining a 1-2 pitch down the right field line for the game's key blow.
Michael Saunders went 3-for-4 with an RBI for the winners.
Yoervis Medina (4-3) got the decision with a scoreless eighth and Danny Farquhar picked up his seventh save as the Mariners took 2-of-3 in the series.
Adrian Beltre and Mitch Moreland each had an RBI for American League West-leading Texas, which came into the game 1 1/2 games ahead of second-place Oakland.