Guillaume Latendresse scored for Minnesota 3:33 into the game, but Sutter tied it on a power play with 10 seconds left in the first period.
Babchuk put Carolina in front at 5:58 of the second period on another power play in which coach Paul Maurice lined up three forwards.
"I'll do it again. If the power play struggles, it affects the offense, and it has been good to think of something which could be positive," Maurice said.
New Minnesota center Matt Cullen tied it at 9:52 against his former team, but Carolina upped the pace and took control.
Jokinen restored the Hurricanes' lead on a two-man advantage at 11:16, and Sutter made it 4-2 at 18:03 after a great rush by Patrick Dwyer up the right wing.
"We were nervous during the first 10 minutes," Jokinen said. "Then we got bigger intensity and energy and played a good second period. They have two good units offensively, but their weakness is the defense, and we used that, putting pressure on their defensemen."
"He had three chances and brought out a great save from the goalie," Maurice said. "He's still learning the game, and the first thing is earning the confidence of the coaches, get us thinking about how many minutes he should have, and if he could make our top nine forwards. If it stays like this, he will. He has eight more games to convince us."
Ward came away with 29 saves, while Minnesota's Niklas Backstrom turned away 31 shots.
The teams play again in Helsinki on Friday.
|Hurricanes captain Eric Staal is taken down as he attempts to score from a tough position. (Getty Images)|
|Carolina (1-0-0) «||1||3||0||4|
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|03:33||Guillaume Latendresse (1), Wrist Shot. Assist: Koivu, Brodziak|
|19:50||Power Play - Brandon Sutter (1), Tip-In. Assist: Jokinen, McBain|
|05:58||Anton Babchuk (1), Wrist Shot. Assist: Cole, Dalpe|
|09:52||Power Play - Matt Cullen (1), Wrist Shot. Assist: Havlat|
|11:16||Power Play - Jussi Jokinen (1), Wrist Shot. Assist: Corvo, Pitkanen|
|18:03||Brandon Sutter (2), Snap Shot. Assist: Dwyer, Gleason|
|16:39||Power Play - Brent Burns (1), Wrist Shot. Assist: Miettinen, Cullen|
|10:38||Zanon, 2 min, Tripping|
|19:15||Barker, 2 min, Interference|
|47.0||Harrison, 2 min, Tripping|
|09:15||Gleason, 2 min, Cross-Checking|
|10:29||Barker, 2 min, Holding|
|10:39||Cullen, 2 min, Slashing|
|12:42||Zanon, 2 min, Hooking|
|51.0||Staal, 2 min, Tripping|
|12:41||Babchuk, 2 min, Hooking|
|15:11||McBain, 2 min, Holding|
|Shots On Goal|
|Attendance - 12355|
|Referees - Tim Peel, Rob Martell|
|Linesmen - Brian Mach, Andy McElman|