Kinsler also had an RBI single and finished with two RBI while Josh Hamilton drove in Texas' other run.
Dustin Nippert (4-1) was the winner, allowing two runs on six hits over six innings. He struck out three and walked three.
Frank Francisco picked up his 17th save.
Junichi Tazawa (1-2) took the loss after giving up four runs on 10 hits in five-plus innings for the Red Sox.
David Ortiz and Dustin Pedroia had solo home runs for Boston, who have lost nine of their last 13 games.
Teagarden's home run off the left field foul pole in the sixth proved gave Texas a 4-2 lead and to be the winning run.