The win snapped a two-game skid and stopped the Yankees from sliding further back in the hunt for the AL wild card, a race in which the Rangers have also become a looming contender.
Ponson (7-2), released by Texas in June, gave up three runs and six hits in 6 1-3 innings. Rivera, who hadn't pitched since Friday because of back spasms, threw a one-hit ninth for his career-best 27th consecutive save this season.
Derek Jeter had two hits and two RBIs after attending a memorial service earlier in the day in Oklahoma City for former Yankees star and longtime broadcaster Bobby Murcer.
The victory was just the fourth in 11 games for the Yankees, who avoided dropping four straight on the road for the first time this season.
New York catcher Ivan Rodriguez, acquired last week from Detroit, left with a bruised right knee in the second inning after Rangers outfielder David Murphy plowed into him at home plate. Rodriguez is day-to-day.
His status was pretty good news for the Yankees, already struggling with injuries late in the season.
New York pitcher Joba Chamberlain was placed on the 15-day disabled list before the game with rotator cuff tendinitis in his right shoulder. The Yankees said Chamberlain will rest for about a week before starting a throwing program.
With Chamberlain's timeline unknown - and Chien-Ming Wang and right-hander Phil Hughes still out - the Yankees were glad to get a solid outing from Ponson, who earned his third win since joining the club in June.
Ponson loaded the bases with no outs in the second inning before Chris Davis' single gave Texas a 2-0 lead.
But the Yankees didn't trail long. Robinson Cano and Jose Molina, who replaced Rodriguez, singled to start the third, and Wilson Betemit followed with a run-scoring double.
The Yankees tacked on two more in the third off starter Tommy Hunter (0-1), who was sloppy in his second career start. The right-hander gave up five runs and eight hits in 4 1-3 innings.
Michael Young homered for the Rangers, who had won three straight.
The Rangers might soon be without Murphy, their standout rookie who was also injured in unsuccessfully trying to barrel over Rodriguez at the plate. Murphy stayed in the game, then left in the fourth with a strained ligament in his right knee.
Texas said Murphy - who leads AL rookies in RBIs, hits and doubles - would have an MRI exam Thursday.
Notes: With Chamberlain on the DL, the Yankees are expected to call up Ian Kennedy from the minors to start Friday. Dan Giese will start Saturday. ... Yankees DH-LF Hideki Matsui hit 37 pitches during batting practice at the club's facility in Tampa, Fla. Out since June 22 with a left knee injury, Matsui is expected to begin a running program Thursday. ... RHP Kevin Millwood, who began the season as Texas' No. 1 starter, had a bullpen session and will make a rehab start for Double-A Frisco on Saturday. Millwood has been on the DL since July 26 with a right groin strain. ... Texas DH Milton Bradley missed his eighth straight start due to a strained left quadriceps but was available to pinch hit. Rangers manager Ron Washington said Bradley could be ready to return to the lineup this weekend at Baltimore.
