With the game tied 3-3 in the seventh, the Rangers put Elvis Andrus on third base with two outs and Berkman sent a 1-0 pitch from Nick Hagadone just inside the left field foul pole for the decisive blow.
Jeff Baker also supplied a two-run shot for American League West-leading Texas, which won a second straight following a three-game skid.
Reliever Robbie Ross (3-1) earned the decision after tossing a scoreless seventh and Joe Nathan racked up his 20th save to preserve the victory.
Cleveland's Scott Kazmir (3-4) surrendered four runs on eight hits over six full innings.
Carlos Santana stroked a bases-clearing double for the Tribe, who dropped an eighth consecutive contest.
Celebrity Families of 2014 [PHOTOS]