Playing against his old club, Rodriguez became just the fourth Major League player to homer in his 40s having also done as a teenager.
It was an 11th home defeat in 12 for the sorry Rangers.
The Chicago Cubs recovered from a 4-0 deficit to beat the Colorado Rockies 9-8.
The Rockies had a four-run ninth, triggered by Carlos Gonzalez’s second homer, but Kris Bryant responded with a two-run walk-off to get the win for the Cubs.
The White Sox of Chicago beat the Red Sox of Boston 10-8 at Fenway Park, Chicago’s Adam Eaton finishing a homer shy of a cycle.
A grand slam in the fourth from Kolten Wong helped the St Louis Cardinals to a 4-1 win over the Cincinnati Reds
The Kansas City Royals opened up a 10-game road trip with a 9-4 win over the Cleveland Indians.
Eric Hosmer scored four runs and Kendrys Morales added three RBIs as the Royals recorded their 60th win of the season.
Matt Wieters was the 11th-inning hero for the Baltimore Oriles as they won a third straight game by beating the Atlanta Braves 2-1.
Wieters hit Luis Avilan over the wall to secure the match – his fourth homer of his career.
Two homers from Curt Casali helped the Tampa Bay Rays to a 5-2 win over the Detroit Tigers and the San Francisco Giants beat the Milwaukee Brewes 4-2.
Elsewhere, the Arizona Diamondbacks beat the Seattle Mariners 4-3.