Richards converted in the second round of the shootout, Raycroft denied all three Carolina shooters, and the Stars beat the Hurricanes 2-1 on Friday.
Richards lifted the puck over Cam Ward's pad after a series of head fakes, and Raycroft stopped Eric Staal's wrist shot in the third round to secure the extra point.
"I went in with speed, slowed down a little bit to see if I could get [Ward] backed up," said Richards, 3 for 5 in shootouts this season. "I wanted to shoot and he didn't give me that, he's a good patient goalie, so I had to think of something quick and I'm lucky I got it up at the end."
The Hurricanes haven't won in Dallas since Feb. 11, 1996, when they were the Hartford Whalers. A group of fans displayed a Whalers flag in a section behind one of the goals.
The Stars are 10-1-1 against the Whalers/Hurricanes since moving to Dallas in 1993.
With No. 1 goalie Lehtonen out with lower back stiffness, Raycroft made 22 saves. Raycroft also is expected to start Saturday night at Phoenix.
"Andrew Raycroft was just outstanding," Stars coach Marc Crawford said. "What can you say about Brad Richards? He just finds a way to score and it's such an important element in today's NHL, that you've got the shootout. Tonight, we only needed the one goal, so that's great when you've got your goaltender playing as well as Andrew Raycroft played for us. It's a huge two points."
Ward made 28 saves and Jiri Tlusty scored for the Hurricanes in regulation.
Dallas is 3-2 in shootouts while Carolina fell to 1-4 in the tiebreaker.
Brenden Morrow nearly won it for Dallas with 3:30 left in overtime, but his rebound shot glanced off the post. Ward came up with a nice stop on Richards with 2:25 remaining in the extra session.
Raycroft made a tough glove save on a shot by Joe Corvo with 42.4 seconds remaining in overtime.
The Hurricanes got little consolation from earning a point. Carolina is 3-5-4 in its last 12 games.
"It's always tough when you lose in a shootout," Ward said. "I thought it was a pretty up-tempo game, both teams were pretty stingy. And then Raycroft makes that great save [on Corvo] in overtime to put it into the shootout, and then the thing about the shootout is it's a flip of a coin. It can go good or it can go bad."
James Neal scored for Dallas at 9:39 of the first period when he took possession to the left of the net, waited for Ward to make the first move, then lifted a shot into the upper left corner.
Tlusty tied it, whipping the puck past Raycroft from the slot at 9:03 of the second period.
The Hurricanes were coming off a five-day break. The Stars are in the midst of a stretch of three games in four nights.
The Stars swept the season series. Dallas won 4-1 at Carolina on Nov. 29.
- Dallas is 9-0-1 in its last 10 at home.
- Lehtonen started 22 of the first 27 games, with Raycroft getting the other five. Lehtonen was limited to 58 games over the previous two seasons due to groin and back problems. Rookie Richard Bachman was Raycroft's backup on Friday night.
- Former Stars RW Jere Lehtinen, the team's last link to the 1999 Stanley Cup champions, officially announced his retirement earlier in the week. He received a standing ovation when he was shown on the big screen during a break in the second period.
- Neal stretched his points streak to six games.
- The attendance was announced at 13,012, but there were half that number in the seats.