Span's bases-clearing double capped a seven-run first inning for the Twins, who scored the most runs ever in a game at Target Field, which opened in 2010.
Ben Revere contributed four hits and three RBI while Justin Morneau went 4-for-5 with a run-scoring double as Minnesota snapped a five-game losing streak.
Scott Diamond (8-3) corralled the win despite surrendering five runs on nine hits over six full frames.
Chris Tillman (1-1) didn't make it out of the first inning for Baltimore, allowing seven runs (only one earned) on five hits in two-thirds of a frame.
Mark Reynolds went 3-for-4 with a three-run homer and four RBI, but his error allowed two runs to score and opened the door to the Twins' big first inning.
Eric Chavez added a two-run shot for the Orioles.