The Marlies' win over the Moose was their first this season in eight games.
John Mitchell, Brett Engelhardt, and Alexander Suglobov also scored for Toronto (33-39-2-5).
Mike Brown and Michael Grabner replied for Manitoba (45-23-7-5).
Ford, playing in his fourth game with the Marlies, was at his best when Manitoba was pressing in the third period.
Dov Grumet-Morris, a college star at Harvard University also making his fourth start, made 16 saves for Manitoba.
The Marlies opened the scoring at 2:29 when Mitchell snapped a wrist shot from a sharp angle and the puck sailed over a surprised Grumet-Morris's blocker.
It took unil the 14:25 mark of the close-checking contest for the Moose to register their first shot on goal, by defenceman Alex Edler when Toronto's Tyson Nash was sent off for hooking. Toronto was assessed all three first-period penalties.
With less than two minutes remaining in the period, Engelhardt, standing directly in front of the crease, re-directed Staffan Kronwall's shot from the left faceoff-circle staking the visitors to a 2-0 lead going into the second.
The Marlies extended their lead to 3-0 midway through the second when Leveille stole the puck from Moose defenceman Dustin Wood at his own blue-line, turned and skated right in on goal beating Grumet-Morris with a high shot.
Leveille, a forward signed as a free agent on April 12th, recorded his first pro goal in his second game with the Marlies. The former college skater spent four years at the University of Maine and was twice voted to the first all-star team.
The Moose finally broke through at 14:53 when Brown outraced defenceman Jay Harrison for the puck and fired the puck on Ford's short side to narrow the gap to 3-1.
Toronto got that one back with about six minutes remaining in the game when Suglobov wristed a rising shot over the Moose netminder's outstretched glove.
With eight seconds remaining in the game, Manitoba's Grabner, the Canucks first-round pick in last year's draft, playing in his second pro game, rounded out the scoring.