Josh Roenicke relieved, only to walk pinch-hitter Chris Gentry, loading the bases.
Brian Duensing then came in to face Martin, who singled to center for the game-winner that broke a tie that had stood since the first inning.
Joe Nathan (5-2) was credited with the decision for American League West-leading Texas, which maintained a two-game lead over second-place Oakland.
Starter Matt Garza held the Twins to one run on three hits through seven innings.
Minnesota counterpart Mike Pelfrey also went seven innings, yielding one run on six hits.
Brian Dozier homered for the Twins.
Celebrity Families of 2014 [PHOTOS]