Richards scored twice within a 3-minute span in the second period to hand the Eastern Conference-leading Rangers a 3-0 advantage and New York went on to post a second straight win.
Marian Gaborik chipped in a goal and an assist, while Carl Hagelin also tallied and added two assists for the victors.
Martin Biron filled in for the flu-ridden Henrik Lundqvist and stopped 27-of-29 shots to earn the win.
Carolina got goals from Tim Brent and Drayson Bowman.
Cam Ward finished with 24 saves for the Hurricanes.