Ryan O'Reilly also scored for Colorado, which has won a team-record eight straight home games. The Avalanche are 9-1 in games that go beyond regulation, including 3-1 in overtime.
"I don't think anybody would have thought that this would happen," Giguere said of the home winning streak. "This is a building that should be hard for opposing teams to play in and we have done it as of late."
Mathieu Garon had 27 saves and Pavel Kubina scored for Tampa Bay. The Lightning are 9-1-3 in overtime and shootouts this season.
"We battled hard. We got a point but we wanted two," Garon said.
Duchene got the winner when he sent a shot off Garon's left pad from the left side with 1:22 left in the extra session for his team-leading 12th goal of the season.
"I tried to fire one quick five-hole," Duchene said. "I think he thought that I was past the net when I fired it. I don't think it was on the right angle but I just threw it and it found its way in. A bit of luck there."
T.J. Galiardi nearly won it late in regulation when he had a short-handed breakaway chance with 1:51 left, but Garon made the save. O'Reilly had a goal waved off with 48 seconds left because he knocked it in with his glove.
Kubina gave Tampa Bay the lead when he scored his second goal of the season with 1:51 left in the first. Kubina picked up a pass from Lecavalier, used Colorado's David Van der Gulik as a screen and wristed a shot through Giguere to make it 1-0.
"We got tons of scoring opportunities and we only got one in," Lightning coach Guy Boucher said. "Sometimes it has a tendency to come back and haunt you."
It did when Colorado tied it midway through the second period. O'Brien took a shot from the point that Garon saved but O'Reilly snagged the rebound and flipped it into the net for his eighth goal of the season.
"That goal was the turning point," Colorado coach Joe Sacco said.
Garon was back in net after being pulled when he allowed three goals in 6:21 against San Jose on Wednesday. He wasn't tested much early with Tampa Bay dominating play at the start. The Lightning outshot Colorado 14-5 in the first and Garon's only test came when he made two saves on the Avalanche's first power play.
The Lightning have struggled on the power play on the road this season and were 0 for 3 Friday. In their last 10 road games, they are 3 for 33 with the man advantage.
"We're losing our games because we can't get that power-play goal," Boucher said. "The power play on the road has been horrendous."
The Avalanche are 13-5-1 in one-goal games. Their 13 wins ties New Jersey for the league lead. ... Colorado rookie Brad Malone is cousins with Lightning LW Ryan Malone. ... The Avalanche/Nordiques franchise has won eight straight home games twice previously, in the 1983-84 and '94-95 seasons when the franchise was in Quebec. ... Lightning RW James Wyman played in his first NHL game since Nov. 28, 2009, when he was with Montreal.
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DENVER, CO - DECEMBER 23: Steven Stamkos #91 of the Tampa Bay Lightning warms up prior to facing the Colorado...