Kinsler, who opened the 10th with a single, was sent home with two outs when a flinch by Danny Farquhar (0-2) was ruled a balk by home plate umpire CB Buknor.
Joe Nathan took the mound for Texas in for the bottom of the 10th and came away with his 38th save. Tanner Scheppers (6-2) pitched the ninth for Texas and earned the win by preserving a 3-3 tie.
Kinsler had four hits for the Rangers, who led 3-0 after the first inning.
Dustin Ackley went 3-for-3 for the Mariners.