Garza was obtained by the Rangers earlier in the week from the Chicago Cubs, with whom he had compiled a 6-1 record. He has allowed no more than two earned runs in any of his last seven starts.
An error by Garza in the sixth inning led to New York's only run. His wild throw to first base allowed Brett Gardner to reach leading off the inning and Gardner eventually scored on a single by Robinson Cano.
Nathan, who suffered his second blown save opportunity in 33 chances on Tuesday against the Yankees, gave up a single to Wells before getting Eduardo Nunez on an infield popup to end the game. It was Nathan's 32nd save.
Andy Pettitte (7-8) suffered the loss, allowing two runs on eight hits in six innings.
2014: The Year in Music [PHOTOS]