New York is 7-1-1 in its past nine games.
"We played well enough to win it," Lundqvist said. "It came down to some extra shots in the end, but we definitely had the looks and we worked hard all over the ice. We just played a really good team. They had a strong game as well. There are a lot of good things to take with us here for next game. I am pretty disappointed right now, but I appreciate the effort from everyone."
Shaw gave Montreal a 1-0 lead at 3:55 of the first period with a wraparound goal. Lindberg tied it at 10:03 by redirecting Jesper Fast's pass to the slot past Price.
Montreal took a 2-1 lead on Weber's power-play goal at 1:42 of the second period.
Nash tied the game 2-2 with a breakaway goal at 9:26. It came 6:05 after he was denied by Price on a breakaway.
Zuccarello scored in the first round of the shootout. Radulov tied it in the second round. Price made saves on Derek Stepan, Jimmy Vesey and Miller to keep the Canadiens in it so Byron could win it.
"All the stuff we've been doing in practice, it's reinforcing everything the coach is telling us to get a win against a very good Rangers hockey team on the road," Byron said. "It can give us a lot of confidence."