The Rangers suffered just their third loss in 17 games.
Cotts (4-2) struck out the first batter he faced, but then walked Nick Franklin and gave up Seager's home run on an 0-2 pitch. Justin Smoak added a home run in the Seattle ninth.
The only Texas run came in the fourth on a single by A.J. Pierzynski.
Seattle starter Hisashi Iwakuma (11-6) lasted seven innings to get the win. He allowed only four hits and struck out eight.
Texas put runners on second and third with one out in the eighth without scoring and Danny Farquhar struck out two in the ninth on the way to his sixth save.