Carolina outshot Tampa Bay 9-5 in the third period.
"The past couple of games we haven't had a good start," Hurricanes defenseman Jaccob Slavin said. "Against a team of that caliber, you have to be able to do it the right way the whole time.
"We've got to get to our game and what makes us successful at the drop of the puck and not wait until it's too late."
Goodrow scored into an empty net with 35 seconds left for the 3-0 final.
"We worked hard; we didn't come out as ready as we needed to in the first period," Brind'Amour said. "That was the game, unfortunately. But we fought back. We might have some off nights, but it's never a lack of effort."