Morales blasted a one-out double into the gap and off the right-center field wall in the 13th against Michael Kirkman (0-1).
Ross Wolf relieved and, after taking third on Carlos Triunfel's groundout, Morales trotted home when Bay poked a broken-bat looper fell in left field.
Morales finished 3-for-6 with a two-run homer for Seattle, which snapped an eight-game losing streak.
Yoervis Medina (1-0) earned his first major-league win with 1 2/3 frames of scoreless relief.
Nelson Cruz had three hits and Jurickson Profar homered for Texas.