Leclaire, who has been out of action with a lower body injury, won for the fourth time in 12 decisions this season.
Kyle Brodziak tipped in a shot 3:36 into the game to put the Wild in front but Leclaire was perfect the rest of the way while Ottawa's special teams rallied them in the second period.
Daniel Alfredsson and Nick Foligno both scored on the power play in the second and Chris Kelly added an empty-net goal with 37 seconds left in the third. Erik Karlsson assisted on all three goals, helping the Senators win for just the fifth time in 16 games.
Minnesota, which has lost seven out of nine, got 22 saves from Jose Theodore.
With the score tied at 1-1, Kyle Brodziak appeared to have given Minnesota the lead. The Wild, however, were called for having too many men on the ice.
It was Ottawa's fifth straight victory over Minnesota.