Pyatt had an open net to shoot at for his seventh goal of the season after Mattias Ohlund had hit the post in a scramble around rookie goalie Mike Smith. The Stars had previously killed 27 consecutive penalties on the road. The teams entered the third period tied 1-1 after Kevin Bieksa scored from centre on Smith and Eric Lindros's point shot beat Vancouver netminder Roberto Luongo.
The victory got the Canucks above .500 at 8-7-1 while Dallas, off to the best start in franchise history and battling Anaheim for top spot in the Western Conference, slipped to 11-3-0.
Luongo gave his club a chance to win with 4:15 remaining when he came out of his net to smother a chance by Mike Modano, who was left alone in front of the Vancouver goalie.
He then gloved the rebound while prone on the ice.
Lindros tied the game after Luongo denied Modano on an earlier break-in. The goal was the fourth of the season and third in three games for Lindros, who signed as a free agent in July.
Earlier, Stu Barnes intercepted Ohlund's outlet pass and skated in on Luongo but drilled the puck off the post.
The Canucks enjoyed a 13-3 shots advantage and a 1-0 first-period lead when Bieksa's unusual second goal of the season came just as a Dallas penalty ended.
He stepped over centre and slapped the puck toward the Stars' goal. The puck appeared to graze defenceman Trevor Daley before entering the net between Smith's pads.
Earlier, the 24-year-old Kingston, Ont., native, making his third NHL start after being selected 161st overall in the 2001 entry draft, almost gave Ryan Kesler a gilt-edged chance when he mishandled a Vancouver shoot-in.
But he prevented Vancouver from taking the lead in the second period when he got his goal stick paddle down in time to foil Alexander Edler on a shorthanded breakaway.
Barnes and Philippe Boucher took delay-of-game penalties two seconds apart in the second period by shooting the puck out of the rink.
However, Canuck Jan Bulis nullified a potential one minute 58-second two-man advantage by taking an offensive zone tripping penalty.
NOTES: With key defencemen Willie Mitchell (concussion) and Sami Salo (knee sprain) still on the shelf, Edler, a 20-year-old rookie, made his second start for the Canucks and earned his first NHL point ... Marc Chouinard sat out a second consecutive game as a healthy scratch for Vancouver while Tommi Santala drew into the lineup ... Vancouver continues its six-game homestand Thursday against the Anaheim Ducks ... Mike Burnstein made his 1,000th appearance as Vancouver's medical trainer ... he's outlasted five head coaches during that span.