Greiss made 29 saves for the Islanders, who are two points ahead of the Columbus Blue Jackets for the first wild card into the Stanley Cup Playoffs from the Eastern Conference.
"I think we got on our heels a bit," New York forward Jordan Eberle said. "We lost some battles, and they're a good team. I guess you can say you got a point, but this late in the season, you want to try and get two. You take the one, but some things to correct, obviously."
The Blues outshot the Islanders 23-5 after the first period.
Dunn tied it with an unassisted goal when he beat Greiss with a one-timer from above the circles after New York's Ryan Pulock backhanded the puck from the right face-off dot.
"I honestly didn't know it went in until I saw everyone celebrating," the defenseman said. "It's great for me, but it's also great for the team."