Martin hit a game-winning bases-loaded single with two outs in the ninth inning to give Rangers a 2-1 victory over the Minnesota Twins.
"I was concentrating on getting a good pitch to hit and trying to get a base hit anywhere to win the game," Martin said. "I was just looking for a good pitch to hit."
"We put ourself in a position to win, and Leonys came through," Manager Ron Washington said. "He stayed short, and hit the ball up the middle."
The Rangers built their winning run against three Minnesota relief pitchers.
Alex Rios doubled on a 3-0 pitch with one out against losing pitcher Caleb Thielbar (2-2). Thielbar then walked pinch hitter Jeff Baker intentionally and retired Jurickson Profar on a fly ball, with Rios taking third base.
"The 3-0 hack by Rios was huge," Washington said. "Gentry fought for that at-bat and got us in position."
With the left-handed-hitting Martin coming up, Twins manager Ron Gardenhire called on left-hander Duensing, against whom opponents had been batting just .136 in his past 11 appearances. Duensing got ahead 1-2 in the count.
Martin, after going hitless in his first four at-bats, came through with the second walkoff hit of his career. The first was a three-run, 10th-inning home run against the Los Angeles Angels July 30.
"I wasn't looking for a homer (Saturday). I was looking to stay within myself, and get a hit anywhere. The key is to win the game," Martin said.
Joe Nathan (5-2) pitched a scoreless ninth for the win. It gave AL West-leading Texas a 20-7 record for August, the league's best.
Brian Dozier led off the game with his 15th home run, a season record for a Minnesota second baseman.
Dozier's homer broke a tie with Tim Teufel (1984) for the most by a Twins second basemen.
Without Morneau, the Twins managed just five hits against Garza and relievers Jason Frasor and Nathan.
Their 11 strikeouts marked the 51st time this season Minnesota batters have struck out at least 10 times.
However, Andrus left the game that inning because of tightness in his lower back. Adam Rosales pinch ran for Andrus and scored on the sacrifice fly.
The Rangers indicated that Andrus' removal was precautionary, and listed him day to day.
Garza threw a season-high 119 pitches and struck out nine, his highest total in eight starts for Texas. He allowed only three hits and walked three in his seven innings.
"Garza threw the living fire out of the ball," Gardenhire said.
"Normally when you give up two runs you win a ballgame. We were really happy with Pelfrey. We thought he was unbelievable. We just couldn't come up with any hits to knock in any runs."
Pelfrey, who did not walk anyone, threw 98 pitches in matching his longest outing this season.
The Rangers managed just six hits, all singles, against him. Pelfrey retired 13 consecutive batters between the third and sixth innings.
The Twins recalled catcher Josmil Pinto from Triple-A Rochester to take Morneau's roster spot. OFAlex Presley, acquired from Pittsburgh in the Morneau trade, will join Minnesota on Sunday. Manager Ron Gardenhire said Presley would play center field and bat leadoff. ... Texas will activate designated hitter Lance Berkman, right-handers Nick Tepesch and Neftali Feliz and left-hander Michael Kirkman from the disabled list, and recall righty Cory Burns and catcher Robinson Chirinos and select outfielder Jim Adduci from Triple-A Round Rock on Sunday. That's when the active roster can expand to as many as 40 players.