Morneau highlighted a four-run first inning with a three-run blast and hit a solo shot in the fifth to give the Twins a 6-4 lead.
Derek Jeter blasted a homer for the Yankees in the ninth with one out, but Matt Capps then struck out Curtis Grandson and got Mark Teixeira on a fly ball to right to end the game and even the series at 1-1.
Jason Marquis (1-0) was brought up from Class AA New Britain prior to the game and picked up the win by going five innings. Marquis allowed four runs on seven hits and survived a three-run first that featured a two-run single by Nick Swisher.
Hiroki Kuroda (1-2) allowed six runs on 10 hits in 4 1/3 to suffer the loss.