"We kept the game close long enough for the offense to do its thing," said Rangers starter Rick Helling (10-7).
The Rangers have won six of seven to move 24 games over .500, the first time they've done that since a club-record 26 above .500 at the end of 1977 (94-68).
Palmeiro and Greer hit solo shots to help the Rangers to a 4-1 lead.
Helling (10-7) held Boston to one run and six hits in six-plus innings. Nomar Garciaparra, the AL's leading hitter, had half the hits, going 3-for-3 with a double.
"It's funny how this game works. The results looked pretty good, but stuff-wise, I didn't feel that great," Helling said.
The Red Sox got to Helling for two doubles and a triple.
"To have the bases loaded with their 3-4-5 hitters coming up, and to get out of it with just one run, that has to be pretty good," Texas manager Johnny Oates said.
"Overall, we have to be very pleased with our bullpen this year. The numbers say they match up with anyone in the league."
In the Texas seventh, reliever Rheal Cormier fumbled Rusty Greer's grounder and Stanley then dropped the throw at first. Juan Gonzalez singled and Palmeiro followed with his 36th homer to make it 7-2.
An inning later, Greer followed with a two-run homer.
"It's the nature of the game to have stretches where we play some good baseball and bad baseball. Right now, we're swinging the bats pretty well," Greer said.
Palmeiro has six homers in his last six games, 11 in August and 350 for his career, passing Albert Belle for 54th on the career list.
Five of Texas' six hits off Red Sox starter Brian Rose (6-5) were for extra bases two homers and three doubles. Greer's leadoff homer in the fifth padded the Rangers' lead to 4-1 and chased Rose.
Palmeiro's solo homer in the third inning put Texas up 3-1.
Greer's second homer of the game was a two-run shot in the eighth off Bryce Florie. He has 15 homers this season.
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