When the final 20-minute period began, Minnesota trailed 2-1.
James Sheppard created a 2-2 tie with his second goal this season at the 2:02 mark.
Koivu's goal at 6:10 gave Minnesota a 3-2 lead, and later, Eric Belanger scored his sixth goal into an empty net (unassisted) at 19:37.
Sheppard also had one assist, Marc-Andre Bergeron had two assists, while net minder Niklas Backstrom made 35 saves on 37 shots for the Wild, who recorded their first win this season when trailing after two periods.
Tampa Bay got goals from Radim Vrbata (power play) in the first period and Ryan Malone in the second.
Net minder Mike Smith surrendered three goals on 32 shots in the Lightning's fourth straight road loss.
On the power play, Minnesota went 0-for-3 and Tampa Bay went 1-for-5.