Crouse cut it to 3-1 at 10:41 with his 20th goal of the season, scoring from the top of the slot five seconds after an Arizona power play expired.
The Hurricanes, who didn't get a power play in a 3-2 loss to the Golden Knights on Wednesday, then scored on three consecutive power plays and ended 3-for-4 with the man advantage.
"Three out of four on the power play, it's just a game changer," Necas said. "It kills any momentum that they get against us."
Gostisbehere pushed it to 4-1 at 12:26 with a wrist shot from the left face-off circle.
Gostisbehere, who had 10 goals in 52 games with Arizona, joined Carolina earlier in the day and went through only a brief morning skate before playing.
Brind'Amour said it's evident Gostisbehere will be a major addition, especially on the power play.
"It was a tough situation, come in and play on a new team and then play your old team," Brind'Amour said. "But he looked really comfortable and didn't look like he was nervous or didn't know what he was doing. Obviously, you can see he's going to help us."