Ryan Kesler put the Canucks ahead in the fourth round, but Mike Modano tied it with a shot that trickled in off Jason LaBarbera's glove. After Marty Turco stopped Alex Burrows' deke, Neal beat LaBarbera with shot past the blocker.
Modano had a goal and an assist, and Mike Ribeiro also scored in regulation as the Stars won despite not arriving in Vancouver until late Sunday morning after their late flight out of Edmonton, where they lost 4-1 Saturday night, was canceled.
Turco gave up two first-period goals a second straight night, but finished with 33 saves through overtime as the Stars improved to 6-2-1 in their last nine games.
Kyle Wellwood and Darcy Hordichuk scored in regulation for the Canucks, who also lost a shootout in Atlanta on Friday night, and are 1-5 in the shootouts this season.
LaBarbera made his third start in four nights -- but first in his hometown -- since being acquired from Los Angeles five nights earlier.
He made six of his 34 saves in overtime, but his best were in the third period.
LaBarbera robbed Steve Ott on a one-timer off the rush 2 minutes into the third, but saved his best stop for a Stars' power play 10 minutes later, diving with his glove on the goal line to take an empty net away from Ribeiro.
Turco answered with a sprawling stop to rob Mason Raymond on an equally empty net after a 2-on-1 feed from Wellwood less than a minute later.
Wellwood opened the scoring 2:11 in with his 14th goal, driving to the net to redirect Steve Bernier s pass out of the corner past a prone Turco. It was a quick response from the Canucks' second line after being called out publicly by coach Alain Vigneault after the 4-3 shootout loss in Atlanta.
Hordichuk doubled the lead on a similar play with 4:20 left in the first, but Modano closed the gap by beating defenseman Mattias Ohlund to the net on a 2-on-2 rush and redirecting Neal's pass behind LaBarbera with 10 seconds left.
Ribeiro tied it shortly after the first penalty -- for too many men on the ice, the Canucks' NHL-worst ninth bench minor -- in a two-man advantage expired, firing a cross-ice pass from Jere Lehtinen past a sprawling LaBarbera.