Rachunek scored one of three power-play goals for the Rangers, who have won three straight.
Matt Cullen, who played for Carolina last season, added two assists for New York. Brendan Shanahan and Pascal Dupuis, in his Rangers debut, also scored.
Andrew Ladd replied for the Hurricanes, who lost all-star winger Justin Williams midway through the second period with what the team called an upper-body injury. Defenceman Bret Hedican also left the game with a concussion.