Al Jefferson finished with 18 points and 12 rebounds, and Rashad McCants scored 18 points for the Timberwolves (2-3), who led by as many as 23 points in the first half.
Jeff Foster had 12 points and 20 rebounds for the Pacers (4-1). Marquis Daniels and Jamaal Tinsley both added 18 points and Travis Diener had 16.
Late in the fourth quarter, Daniels' runner gave the Pacers a 92-89 lead, but Jefferson's three-point play sent the game into overtime.
Shawne Williams' free throw put Indiana ahead 104-102 with 14 seconds remaining in the first overtime, but Minnesota's Chris Richard sent the game to a second extra period by tipping in Davis' missed jumper at the buzzer.
Trailing 110-109 in the second overtime, the Timberwolves went back in front on Smith's two free throws with 1:25 left. A free throw by Greg Buckner extended the lead to two points with 14 seconds remaining, and Davis made it 113-110 with a free throw seconds later.
The Pacers had a chance to tie, but Diener's 3-pointer was short with time winding down.
Pacers forward Mike Dunleavy left the game in the third quarter after getting poked in the right eye. He did not return. The Pacers were without forwards Jermaine O'Neal (hyperextended left knee) and Troy Murphy (strained left Achilles'). O'Neal is expected to miss Saturday's game at Chicago. Murphy could be out 10 to 14 days.
Minnesota was without guards Randy Foye (left knee tendinitis) and Sebastian Telfair (ankle).