Given little chance to win the series against a team that was 25 points better during the season, Carolina continued its remarkable playoff success in overtime.
Jeff O'Neill, who scored the tying goal in the final seconds of the second period, slid a pass in front of the net from along the goal line. Francis was unchecked at the top of the crease and deflected the puck over the right shoulder of goaltender Dominik Hasek for his sixth postseason goal and second career overtime winner.
The Hurricanes improved to 7-1 in overtime in this year's playoffs and stole home-ice advantage from the Red Wings, who captured the Presidents' Trophy and were expected to have little trouble with a team described often as underdogs.
Carolina also improved to 12-0 in the postseason when scoring at least two goals.
O'Neill helped get the Hurricanes to overtime by beating the end of the second period by 10 seconds. Former Red Wings defenseman Aaron Ward sent him in alone with a pass from just outside his own blue line.
O'Neill got behind Detroit's Sergei Fedorov, who had broken his stick and was heading to the bench. O'Neill bore down on Hasek and slid a shot under the goaltender that trickled toward the goal line. Hasek sprawled to stop it but ended up nudging in the puck with his outstretched left arm.
The Red Wings had four power plays in the game's first 21 minutes to just one for the Hurricanes. But Detroit teammates Igor Larionov and Kris Draper went to the penalty box 37 seconds apart early in the second period, giving Carolina a prolonged two-man advantage.
Defenseman Sean Hill cashed in with 37 seconds left on Larionov's high-sticking penalty, getting a cross-ice feed from Sami Kapanen at the top of the left faceoff circle and slapping a one-timer into the top right corner of the net.
All four of Hill's playoff goals have come on the power play.
The Red Wings squandered two more power plays before the "Grind Line" put them back in front just over seven minutes later.
Draper checked defenseman Niclas Wallin behind the net, dislodging the puck. Darren McCarty got it and passed to the top of the right circle, where Kirk Maltby unleashed a wrist shot against the grain and under Irbe's right arm for his second postseason goal.
It was the first 5-on-5 tally allowed by Carolina since Game Six of the Eastern Conference semifinals against Montreal.
O'Neill shifted the momentum 50 seconds before the end of the period with his fifth goal in the last eight games.
It ended two periods played at breakneck speed and signaled the start of a more tentative and territorial game.
The Hurricanes had two power plays in the third period but combined for just one shot. And it took a clutch play by defenseman Marek Malik to keep the game tied.
With 9 1/2 minutes to go in regulation, Detroit's Brett Hull deked around a defender on the left side, then walked out in front and was poised to shoot on out-of-position goaltender Arturs Irbe. Malik got enough of Hull's attempt to send it just wide of the right goalpost.
Carolina's Sami Kapanen, a 27-goal scorer during the season who has just one in 18 playoff games, never got off a shot on a 2-on-1 with 3:11 left.
The Red Wings had a last chance to avoid overtime when Erik Cole was penalized with 1:41 remaining after twirling fellow rookie Pavel Datsyuk to the ice. But they could not convert and fell to 1-for-7 on the power play.
Detroit was on its third power play when it grabbed the lead 15:21 into the first period.
Tomas Holmstrom drew an interference penalty on Carolina defenseman Glen Wesley at 14:56, then made the key play 25 seconds later to set up the series' first goal. The 6-foot, 205-pound Swede camped in front of the net and was sent sprawling into the 5-8 Irbe by Ward.
That left Irbe helpless to stop Fedorov, who scored his fifth playoff goal off a rebound of Steve Yzerman's shot.