Columbus responded with two goals in a 1:39 span in the second period.
Bemstrom cut it to 2-1 at 8:47, redirecting a backhand centering pass from Eric Robinson at the top of the crease, and Danforth tied it 2-2 at 10:26 on a rebound in front.
"The mindset with this group, no matter how many goals we're down, especially since the All-Star break, the way we've been playing, we're not out of any game," Blue Jackets defenseman Andrew Peeke said. "We've found ways to come back in games, get two points, get a point. But tonight, we couldn't get it done. Obviously, it's not ideal starting down 2-0, so hopefully the next game (at the Boston Bruins on Saturday), we can start with the lead."
Kyle Palmieri put New York back in front 3-2 at 3:16 of the third period, scoring over Merzlikins' left shoulder with a shot from low in the right circle. It was his first goal in eight games.