Leafs forward Bates Battaglia sent a pass from behind the net out to Kilger, who was off-balance when he flipped the puck past goalie Alex Auld at 12:03 of the third period.
Alex Steen, playing his first professional game in the city where he was born while his father Thomas was playing for the Jets, scored the first goal of the game for the Maple Leafs (1-1). The Jets left Winnipeg for Phoenix after the 1995-96 season.
Martin Hanzal tied the game at 8:09 of the second period on a deflected pass that got past goalie Andrew Raycroft. Hanzal was Phoenix's first-round draft pick, 17th overall, in 2005.
Four minutes later, Daniel Winnik gave the Coyotes (1-1-1) a 2-1 lead on a wrist shot that looked like it hit the cross bar and was put in by Hanzal. But it was ruled Winnik's shot had hit the inside of the net.
Jason Blake, who signed a five-year deal with Toronto in July, tied the game five minutes into the third period.