Correia (3-6) scattered four hits, issued one walk and struck out two en route to his first victory at PNC Park since June 22, 2011.
Tony Watson relieved with one out in the sixth and worked out of a jam to preserve a 2-0 lead the Pirates held at the time.
Andrew McCutchen went 3-for-4 with a two-run homer while Josh Harrison added three hits -- including a solo shot -- and scored three times for Pittsburgh, which took the first of a three-game set.
Twins starter Scott Diamond (5-3) yielded four runs on eight hits over 5 1/3 innings.
Denard Span cracked a two-run homer for Minnesota.