Zuccarello would answer with a breakaway goal of his own at 12:17 when Ryan McDonagh hit him with a home run pass at Carolina's blueline, and the crafty winger beat Ward five-hole for his fourth goal of the season that tied it 2-2.
But Bickell's goal on the power play with under one second remaining in the second period proved to be the difference.
Lundqvist finished the night with 18 saves in the loss.
The Rangers were without forward Chris Kreider, who missed his third consecutive game with neck spasms.
New York returns home Sunday night to host the Tampa Bay Lightning at Madison Square Garden.