Correia (3-1) scattered six hits, allowed only one walk and struck out two in the 102-pitch gem that lowered his earned-run average to 2.33 in five starts.
The right-hander turned in his fifth consecutive start in which he has pitched at least seven innings for Minnesota, staking the Twins to a four-game series split with the Rangers.
Minnesota was clinging to a 1-0 lead in the sixth when Rangers starter Alexi Ogando issued a two-out walk to Josh Willingham and Morneau then pounded the first pitch he saw into the right-field bleachers for a two-run shot.
Brian Dozier went 2-for-4 with an RBI for the Twins.
Ogando (2-2) ended up allowing three runs on four hits over six frames.