Begin snapped a 3-3 tie with 9:30 remaining. After Chris Butler's slap shot from the left point was blocked in front, Begin outbattled two Stars defenders to get to the loose puck and fire a wrister that beat goaltender Richard Bachman.
It was the Flames' second victory over the Stars in five days, having won 7-4 in Calgary on Wednesday.
Michael Cammalleri also scored for Calgary, while Joey MacDonald made 27 saves for his first win while making his first start and second appearance overall for the Flames since he was claimed off waivers from Detroit on Feb. 11.
Antoine Roussel, Reilly Smith and Ryan Garbutt scored second-period goals for Dallas, while Bachman stopped 22 shots four days after he gave up six goals on 22 shots in Calgary.
The result was particularly frustrating for the Stars because they reversed a disturbing trend that has plagued them this season by taking control of the contest with a strong second period, scoring three goals while outshooting Calgary 22-12, establishing a new season high for shots in a period. Over their first 14 contests, Dallas had been outscored 19-9 in middle frame, but the tide began to turn Friday in Vancouver when they outgunned the Canucks 3-1 in the second.
With the Stars up 3-1 early in the final period, the Flames pulled even with two goals within a 45-second stretch that quieted the 17,340 in attendance.
First, Horak scored at 1:39 of the third, banging home a point-blank rebound after Bachman made a pad save on Jay Bouwmeester's wrist shot.
Stajan tied it at 2:24 after Lee Stempniak fed a backhand pass from behind the net out front, where Stajan drilled a one-timer past Bachman's glove.
Trailing 1-0 early in the second period, the Stars struck twice within a 56-second span to surge ahead 2-1.
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Roussel tied it just 1:51 in, camping out in front to screen MacDonald and re-directing Cody Eakin's high, floating wrister over MacDonald's glove. The goal was originally credited to Eakin, but replays clearly showed the puck glancing off Roussel for his third of the year and second in the last two games.
Smith then recorded his second goal in as many games when he corralled his own rebound and lifted a backhander from the edge of the crease over a prone MacDonald at 2:47.
The Stars made it 3-1 with 4:37 left in the second on Garbutt's third of the year. MacDonald made a pad save on Stephane Robidas' wrist shot from the left circle, but Garbutt pounced on the rebound and popped it over MacDonald's pad.
Garbutt, who had a fight in the first period, now has three points (two goals, one assist) in four games since he sat out three in a row as a healthy scratch.
Calgary jumped on top 1-0 at 8:11 of the opening period when Cammalleri notched his fourth goal of the season
-- all have come against Dallas. With the Flames still on a power play, and just 6 seconds after a 5-on-3 advantage expired, Cammalleri fired a wrist shot from a sharp angle to the right of the net and it slipped under Bachman's blocker and went in.
NOTES: Dallas D Trevor Daley left the contest after the second period with an upper body injury and did not return. His status was unclear. ... Cammalleri continued his recent success against Dallas. He had a hat trick against the Stars in the Flames' 7-4 win Wednesday, now has 11 points (seven goals, four assists) in the last five meetings between the clubs. Overall, since missing three games with a lower body injury, he has collected four goals and two assists in the last three contests. ... Stars G Kari Lehtonen was placed on the injured list Sunday, retroactive to Friday, with a groin injury. He had to come out of the Stars' 4-3 win over Vancouver. He is not expected to be out more than the week he is required to spend on injured reserve.
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