Haydn Fleury scored his first NHL goal in his 96th game, and Erik Haula scored for the Hurricanes shortly before leaving following a hit from Ducks defenseman Josh Manson.
Petr Mrazek allowed four goals on 13 shots in 25:51 before he was replaced by James Reimer, who made seven saves for the Hurricanes (6-3-0).
Carolina has lost two in a row following a 6-1-0 start.
"I think the boys, hopefully, will learn a lesson from that one," Hurricanes forward Jordan Staal said. "It's not easy. We're showing up thinking that it's just going to be easy. It's not. It's a good league and they played well."
The Ducks scored the first four goals.
Henrique made it 1-0 at 4:06 of the first period. Ondrej Kase fed the puck to Rakell in the middle and he passed it to Henrique, who was wide open coming down the left side. Henrique's shot from the face-off circle beat Mrazek to the short side.
Mrazek made a right-pad save on Manson seven seconds after the Ducks killed the first penalty of the game, but Terry scored 11 seconds later when he deflected a point shot from defenseman Cam Fowler for a 2-0 lead at 13:44.
"Rather than just walking in the room and saying, 'Hey, let's have a good start,' (we gave) them some tangible things to look for there," Eakins said. "Once again they took it to heart, and we were able to get what we needed."
Silfverberg made it 3-0 at 37 seconds of the second period. Michael Del Zotto made a diagonal pass through the neutral zone that made it to Silfverberg, sending him in alone against Mrazek. The goalie made the initial save, but Silfverberg pushed in the rebound.
Rowney scored from a sharp angle to make it 4-0 at 5:56 and bring Reimer off the bench.
"It's not his fault," Hurricanes coach Rod Brind'Amour said of Mrazek. "We left him out to dry. We had a lot of guys sleeping tonight."
Haula scored his seventh goal of the season on a deflection while on the power play at 8:47 to cut it to 4-1, and Fleury scored on a point shot at 12:20 to make it 4-2.
"All the guys have been joking, saying 'It's coming, it's coming,'" Fleury said of his first NHL goal. "This year, I felt from the first game that I've had chances to score each night."
