Richards, Briere and Jeff Carter scored in regulation for the Flyers, and Michael Leighton stopped his former Carolina teammates cold in the shootout.
The Flyers blew a three-goal lead with 11 minutes left but regrouped to improve to 3-2 in shootouts this season. They won consecutive games for the first time in more than a month and kept the Hurricanes winless in three straight.
Jussi Jokinen scored two goals and Brandon Sutter added one for Carolina, which came up empty on attempts by Tuomo Ruutu and Jokinen in the shootout. Richards ended it in the second round by tucking the puck between Cam Ward's legs.