Cliff Pennington and Ryan Sweeney drew back-to-back walks from reliever Warner Madrigal (0-2) to open the eighth. Two outs later, Cunningham hit the first pitch over the wall in left field.
Cunningham homered in his 46th at-bat since making his big league debut this year. He hit 17 home runs combined in Double-A and Triple-A this season.
Joey Devine (6-1) pitched one scoreless inning for the win. Brad Ziegler worked the ninth for his ninth save in as many chances.
Ziegler allowed a two-out double to Marlon Byrd but got Nelson Cruz to ground out to end the game. Cruz drove in all four Texas runs with a two-run double in the third, an RBI double in the fifth and a single in the seventh.
Texas starter Vincente Padilla scattered five hits over five innings in his second start since coming off the disabled list, but was denied his 11th road win of the season. He allowed two runs, struck out four and left with a 3-2 lead before the Rangers bullpen let it slip away.
Reliever Bill White walked the only two batters he faced in the sixth, and after Rajai Davis hit into a fielder's choice, Pennington walked to load the bases. Sweeney then hit a sharp single to the gap in right-center to score Jack Hannahan and Davis to give Oakland a 4-3 lead.
Dana Eveland allowed three runs over five innings but remained winless in his last four starts.
Huston Street, the former Oakland closer who has been relegated to middle relief, gave up the tying run in the seventh when Michael Young singled and scored on Cruz's third hit of the game to make it 4-all.
Rangers designated hitter Gerald Laird was ejected in the fourth inning after being called out on strikes. He disputed the third strike and was walking back to the dugout when he was ejected by home plate umpire Angel Campos.
It was the second time in two days that Campos ejected a Texas player. Milton Bradley was thrown out of Saturday's 7-1 loss after arguing with Campos about a double play.
Notes:@ Rangers CF Josh Hamilton went into the game batting .472 in September but was hitless in five at-bats.