Inserted for the injured Randy McKay, Corkum beat Roy on a breakaway 14:29 into the first period to end the goaltender's Stanley Cup Finals shutout streak at 227 minutes, 41 seconds, 1:41 shy of Clint Benedict's record.
It was Corkum's first postseason goal since Game One of the 1998 Western Conference quarterfinals as a member of the Phoenix Coyotes.
"It is a dream come true, really," said Corkum, who was acquired from Los Angeles at the end of February. "I know we still got a lot of hockey to play, but just to be a part and contribute in a Stanley Cup Finals game is what I have been playing for for a long time."
Just under three minutes later, Stevenson got his first goal since Game 3 of the 1998 Eastern Conference semifinals to give the defending champions their first lead of the series.
"That's the biggest so far," Stevenson said of his game-winner.
Colorado had trailed just 52 seconds in its previous eight contests and had not allowed more than one goal in the first period in any playoff contest.
Martin Brodeur shook off another early goal by Joe Sakic and stopped 19 shots as New Jersey took home-ice advantage from the Presidents' Trophy winners. The Devils, who have won 17 of their last 22 playoff road games, host Games Three and Four on Thursday and Saturday.
Brodeur took a page from Roy's book in the final minute when he gloved Alex Tanguay's bad-angle shot and held the puck aloft.
"Somebody was on me and I didn't have a stick. It was on one of my knees and he was going high and I had to grab it that way," said Brodeur, who grew up idolizing Roy. "I probably left it there because I wanted to make a point. They were taking liberties on me and going to the net real hard."
In the opening minute of the first period, Brodeur was bowled over by Dan Hinote, although no penalty was called. Midway through the third, he stopped Sakic's wrister from the right faceoff circle, a spot from which Sakic beat him in Game One.
"Marty came up with a huge save," Devils defenseman Ken Daneyko said. "It looked like (Sakic) wanted to go to the same spot."
New Jersey benefited from the extra off day after appearing lethargic Saturday night in a 5-0 loss to open the series. But Colorado was the more sluggish team Tuesday.
"You saw everyone paying the price," Daneyko said. "We took the body a lot more, we got our noses dirty, and you have to do that against a great team like Colorado."
"We have to give them lots of credit," Avalanche coach Bob Hartley said. "They played a much better game. I felt that they beat us to the puck quite a few times. Obviously, those two goals gave them life and we had to play catch-up hockey, and that's where they are at their best. We tried, but I felt our pressure on the puck was not good enough."
The game could not have gotten off to a worse start for the Devils. They took a series of undisciplined penalties, the second of which set up Sakic's third goal of the series and league-leading 12th of the playoffs.
Checked hard into the boards by Shjon Podein, New Jersey's Sergei Brylin retaliated and was penalized for interference at 4:53 of the first period. Just over a minute later, Sakic caught Brodeur out of position and scored off a wild goalmouth scramble after Milan Hejduk chipped a backhander off the crossbar.
Sakic scored on Colorado's sixth shot, but the Avalanche had just two more the rest of the period as the Devils rallied.
"We weren't that sharp," Sakic said. "We got one early but after that we should generate a little bit more. It's something we will look at and see what we can do better."
Patrik Elias delivered a retaliatory slash to Chris Dingman at 12:28, giving Colorado its third power play. But defenseman Adam Foote could not keep the puck in at the left point and Corkum's breakaway goal tied it, one second after the penalty to Elias expired.
Corkum had not played since Game Two of the conference finals but bore down on Roy and wristed the puck between his pads for his seventh career playoff goal.
"I didn't know what the shutout record was. It didn't mean anything to me," said Roy, who also had a nine-game Stanley Cup Finals winning streak snapped. "What means (something) to me is winning at the end."
"You can't afford to take the unnecessary penalties that we took and continually expect to be in games," New Jersey coach Larry Robinson said. "I think the thing that won the game for us tonight was our penalty-killing, without a doubt."
The momentum shifted completely to New Jersey when Eric Messier was penalized for roughing with 5:14 to go in the period and Foote joined him in the box 31 seconds later for holding Bobby Holik's stick.
The Devils could not convert the ensuing two-man advantage but grabbed the lead three seconds after Foote's penalty expired. Roy stopped Scott Niedermayer's wrist shot from the blue line, but Stevenson got the rebound and swept a backhander between defenseman Jon Klemm's legs and over Roy's left shoulder.
"The first (shot), Scotty hit me in the wrists. I don't know how it came back to him, but their 'D' stepped in front of it," Stevenson said. "The puck dropped right in front of me and I took a whack at it and it went in. I was just trying to get some traffic in front of Patrick."
Ironically, Stevenson and Roy were teammates on the Canadiens from 1992-95.
"I honestly thought the second goal would not make the difference in the game," Roy said. "That's a bit of a surprise that it did make the difference."