The Rangers' Kyle McClellan (0-1) hit Todd Frazier with a pitch to begin the 11th inning of a 4-4 tie game, and Mesoraco, after failing to lay down a bunt earlier in the at-bat, followed with a mammoth homer down the left-field line for the decisive blow.
Shin-Soo Choo went 3-for-4 with a solo homer while Zack Cozart and Xavier Paul each added an RBI for Cincinnati, which evened its three-game interleague set.
J.J. Hoover (1-5) earned the victory after notching the final two outs of the 10th and Aroldis Chapman recorded his 20th save.
Nelson Cruz went 2-for-4 with an RBI for American League West-leading Texas.
'Duck Dynasty's Phil Robertson on indefinite hiatus after anti-gay remarks
Nativity scenes to be moved at Guantanamo after troops' protest