The Hurricanes failed in their bid to win back-to-back games for the first time in a month. They have only 11 victories in 42 games and are in last place in the NHL with 29 points.
Nashville has won four out of five.
Hornqvist put Nashville in front for good at 2-1 by scoring on the power play with 14 seconds left in the opening period. His second goal also came with the man advantage 11:16 into the third.
Rinne won his third straight start, giving up a goal to Rod Brind'Amour 2:52 into the game and a shorthanded tally to Tom Kostopoulos with 7:57 remaining in the final period.
Cam Ward allowed all four goals on 34 shots.