An RBI single in the top of the second, Suzuki's lone hit, made him the first player in major league history to compile nine consecutive 200-hit seasons, passing the former all-time mark of eight held by Willie Keeler since 1901.
Winner Felix Hernandez (15-5) surrendered four hits in seven innings with a walk and five strikeouts.
Mark Lowe and closer David Aardsma pitched one inning each.
Julio Borbon had three of the six hits for the Rangers, who trail the AL West-leading Los Angeles Angels by six games.
The losing pitcher was starter Derek Holland (7-11).
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