Begin and Morrison scored their first goals for Dallas, and the Stars ended a six-game home losing streak with a 3-2 victory over the Carolina Hurricanes.
"We're depleted so we're asking a lot of the guys who are left," said Stars center Mike Modano, who assisted on Morrison's goal. "It's critical that we get scoring from everybody."
Begin had gone seven games without a point since he was acquired in a trade with Montreal on Feb. 26.
"It feels very good to get it out of the way," Begin said. "Now I can think about something else."
Morrison hadn't scored in four games after being picked up off waivers from Anaheim on March 4.
"I was brought in here to provide offense and when you don't get on the board right away, it's frustrating," Morrison said. "Now it's a matter of establishing some consistency."
Mike Ribeiro, Begin and Morrison scored during the first 9:51 of the second period after Rod Brind'Amour's short-handed tally put the Hurricanes in front in the first.
Ray Whitney's power-play goal from the right circle at 13:45 of the third period -- his 22nd of the season -- pulled Carolina to 3-2.
Marty Turco, starting for the 37th time in Dallas' last 38 games, made 29 saves. Turco had luck on his side when the Hurricanes hit the post twice with shots early in the final period, and Turco withstood a late Carolina flurry for his 31st victory of the season.
The Stars, involved in a nine-team scramble for the bottom four playoff spots in the Western Conference, were coming off a flat performance Tuesday night in a 5-2 loss at St. Louis.
"Those were desperate points and we needed them," Stars coach Dave Tippett said. "When new guys like that help the cause, it endears them in the locker room. We didn't make it easy on ourselves at the end, but we'll take the points."
Dallas' home skid matched the longest in franchise history, with the Stars managing only seven goals during that span.
Carolina's Eric Staal continued his recent hot streak with an assist, giving him 11 points in his last six games.
The Hurricanes lost for the second time in 24 hours. Carolina had its four-game winning streak snapped on Wednesday night, dropping a road shootout to the Chicago Blackhawks, 3-2.
"We looked like a team that was playing our fifth [game] in seven [nights] and got in at 3 in the morning," Carolina coach Paul Maurice said. "We had trouble moving the puck out of our own end."
Carolina, locked in a fight of its own for playoff position in the Eastern Conference, is 8-3-1 in its last 12.
Dallas was on the game's first power play but the Stars got caught deep in the Carolina zone, leading to a 2-on-1 breakout by the Hurricanes. Brind'Amour converted Staal's pass to give the Hurricanes the lead at 7:23 of the opening period. It was Brind'Amour's 28th career short-handed goal.
"I thought we came out pretty good in the first period," Staal said. "We got off to a one-goal lead, which is what we wanted to do, and we had some chances in the third period. We hit a couple of posts and had a couple of other good looks. It was a big game and we didn't come out ahead."
The Stars got a break at 1:01 of the second period, drawing even at 1 when Ribeiro's pass glanced off the skate of Hurricanes' defenseman Tim Gleason and past Ward.
Dallas took a 2-1 lead when Begin set up in the crease and knocked a loose puck by Ward at 6:35 of the second period, and Morrison's deflection 3:26 later gave the Stars a two-goal pad.
The teams hadn't met in Dallas since Dec. 2, 2005, a 5-4 win by the Stars. The Hurricanes are 0-6-1 in Dallas since moving from Hartford in 1997. ... Carolina D Joni Pitkanen sustained a lower-body injury in the game at Chicago and was scratched against Dallas. ... The Stars' other six-game home losing streak was in 1970 when the franchise was based in Minnesota.
Copyright by STATS LLC and The Associated Press
Arena: American Airlines Center Location: Dallas, Texas
Referees: Rob Martell, Bill McCreary Linesmen: Brad Lazarowich, Don Henderson
Attendance: 16,788 (90.6% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Rod Brind'Amour (12)
(Shorthanded) Assists: Eric Staal, Cam Ward
Mike Ribeiro (19)
Assists: Matt Niskanen, Loui Eriksson
Steve Begin (7)
Assist: Chris Conner
Brendan Morrison (11)
Assists: Mike Modano, Nicklas Grossmann
Ray Whitney (22)
(Power Play) Assists: Rod Brind'Amour, Matt Cullen