Murray, who was acquired in a trade with the Ottawa Senators on July 11, played the first two periods. He is competing for Toronto's No. 1 job with Ilya Samsonov, who also made 16 saves on 16 shots in two periods against the Senators on Saturday.
Denis Malgin had a goal and an assist, and Erik Kallgren made 11 saves to finish the shutout for Toronto.
Sam Montembeault started and made 14 saves in 31:10 for Montreal. He was replaced by Kevin Poulin, who allowed two goals on 14 shots in relief.
Nicholas Robertson gave the Maple Leafs a 1-0 lead at 6:53 of the first period. Murray had the secondary assist on the play.
Malgin made it 2-0 with a power-play goal at 19:15 of the second period, and Nick Abruzzese scored another goal on the power play at 16:35 of the third period for the 3-0 final.
