Panarin took a shot from above the right circle that went wide, kicked off the end boards and came out the other side, where Fox backhanded the puck into the net for a 1-0 lead at 3:20 of the first period.
The Ducks, who had one short-handed goal through the first 70 games, scored short-handed for the second consecutive game to tie it 1-1 at 16:55.
Carlsson broke up a pass in the New York zone and then fed the puck ahead to Alex Killorn for a partial breakaway and he scored his own rebound.
The Rangers went 1-for-7 on the power play and did not capitalize on a 5-on-3 for 1:36 in the third period.
"Our (power play) unit basically gave them the game tonight," J.T. Miller said. "They scored, basically, twice against us."
Carlsson, who only recently started killing penalties, scored short-handed off a feed from Killorn in a 6-2 win against the visiting Boston Bruins on Wednesday.
"Fun to get on the board two games in a row," Carlsson said. "Feels good out there. It's not too hard, it's not too easy either, so it's fun."