The Twins won for the third time in five games this season despite letting a three-run lead get away.
Minnesota's final run was unearned due to a fielding error by first baseman Chris Davis that allowed Hicks to reach.
Hicks eventually moved to second on a walk to Josh Willingham and came home when Morneau, who had a three-hit day, singled to center with two outs. The winning run came off Jim Johnson (0-1), who came into the game in the ninth inning.
Josh Roenicke (1-0) picked up the victory by working three innings of one-hit, scoreless relief. Glen Perkins turned in a perfect ninth for his first save.
Chris Parmelee's two-run homer highlighted a four-run third that gave Minnesota a 5-2 lead.
A two-run single by Adam Jones brought Baltimore back into a 5-5 tie in the fifth and that was the way it stayed until Morneau's key hit in the ninth.