Modano scored on a power play with 2:21 remaining in the Stars' 4-3 victory over the Oilers on Sunday night.
Modano broke a tie with his 537th career goal, beating rookie Jeff Drouin-Deslauriers from the left circle off a feed from Brad Richards to become the 31st NHL player with 1,300 points.
"That was a great shot there, especially on a milestone goal that was so meaningful for our group," Stars coach Dave Tippett said. "You don't get 1,300 points without being a great player, and that was a great goal."
Modano also had an assist and extended his lead among U.S.-born goal-scorers.
"[Richards] made a great pass, on the stick, and I found a seam," said Modano, playing in his 20th season -- all with the Stars. "I've been blessed to have been around this long."
Toby Petersen scored his first goal in two seasons, and Mike Ribeiro added a power-play goal for Dallas. The Stars improved their regular-season home record against Edmonton to 23-4-2 since they moved from Minnesota in 1993.
The honors continue for Modano, the franchise's greatest player.
"When you think about all he's done, the milestones and the ceremonies, and one more creeps up on him," Stars goaltender Marty Turco said. "At this juncture of the season we need him more than anyone."
Modano, second on the team with nine goals, wasn't sure he would be healthy enough to play after overnight back spasms. But a pep talk from his father, Mike Sr., visiting for the holidays, convinced Modano to lace on the skates.
"It spasmed up in the middle of the night, but it felt better as the day went on," Modano said. "Dad gave me a little lip service, told me not to sit this one out."
Oilers coach Craig MacTavish said leaving Modano alone on a late-game power play was a major tactical error.
"It was missed execution on a pretty elementary part of our penalty kill," MacTavish said. "The game was there for us. It was ripe for opportunity. We squandered a pretty good second and third [periods] with boneheaded mistakes."
Sean Avery gave Dallas -- last in the Western Conference -- a 3-2 lead with 5:09 remaining off a centering pass from rookie James Neal, but the Oilers tied it with 3:22 left on Kyle Brodziak's goal off a rebound, his second goal of the game.
Ales Hemsky also scored for Edmonton. The Oilers were 4-2 winners at St. Louis on Saturday night, but Edmonton is 2-5-1 in its last eight.
Ribeiro put the Stars in front at 7:20 of the first period. Modano's shot from the right point was stopped by Drouin-Deslauriers, but the puck popped into the air, rolled down the goalie's arm and into the crease, where Ribeiro nudged it over the goal line.
"It was a bad bounce," said Drouin-Deslauriers, who hadn't played since Nov. 13. "I batted the puck in the air. I needed to back up in my net. It was bad bounce off my pads. When you're lucky, that hits the top of the net."
Petersen's tip-in of Matt Niskanen's shot pushed Dallas' edge to 2-0 at 14:01 of the opening period. The goal was Petersen's first for the Stars and first in the NHL since March 27, 2007, while a member of the Oilers.
Hemsky's power-play goal from the left circle at 12:11 of the second cut it to 2-1, and Brodziak's rebound goal tied it at 2 at 4:21 of the third.
Losing two leads in the third period caused doubt to creep in for the stumbling Stars, 3-5-2 in their last 10 games.
"It's a pattern, but finally our power play was something to get excited about," Modano said. "It finally won us a game. It's been a while. It's been a real struggle at times, but you've got to brush yourself off and come to work."
Among the missing for the injury-riddled Stars were D Sergei Zubov (lower body), C Steve Ott (hand) and RW Jere Lehtinen (upper body). Zubov will be out for another week, said Tippett, who would not elaborate on the nature of the injury. ... Dallas acquired D Andrew Hutchinson in a trade with Tampa Bay earlier in the day. He will be with the Stars on Tuesday night in Calgary. The Stars sent minor league RW Lauri Tukonen to the Lightning. ... Hemsky has 18 points in his last 15 games.