Steve Ott carried the puck deep in the Vancouver end and passed to Trevor Daley at the left circle. Eriksson skated to the net and fired Daley's goal-mouth pass by Roberto Luongo.
Mike Ribeiro scored the tying goal for Dallas with 52.5 seconds left in regulation.
With goalie Kari Lehtonen on the bench in favor of a sixth skater, Ribeiro deflected Alex Goligoski's shot from the point for his 14th of the season.
Michael Ryder netted a power-play goal for the Stars, who hadn't won four in a row since Oct. 29 to Nov. 8.
Ryan Kesler scored in the opening minute of the second period for the Canucks, who were 8-1-1 in their previous 10 road games.
Mason Raymond added a goal and Chris Higgins had two assists for Vancouver, which improved to an NHL-best 87 points. The Canucks were 12-1-3 in their previous 16 games overall.
Dallas has 70 points, eighth in the Western Conference and one point behind seventh-place San Jose.
The teams played the first of four meetings this season. Last season, Vancouver swept the four-game series by a combined score of 20-5.
Vancouver held an 8-6 shots advantage in a tight-checking, scoreless first period.
The Canucks picked up the pace at the start of the second and were rewarded when Kesler notched his 19th goal of the season.
Kesler's shot from the top of the left circle came off a 3-on-2 break 56 seconds into the period, glancing into the net off Lehtonen's glove.
Lehtonen allowed another shaky goal when Raymond's sharp-angle shot from the left boards went in between the goalie and the right post at 13:11 of the second. Raymond's eighth of the season gave the Canucks a two-goal lead.
Ryder scored his team-high 25th of the season on Dallas' second power play of the game with a one-timer from the left circle at 18:04 of the second.
NOTES: Dallas rookie D Philip Larsen was in the lineup despite a cut on top of his head sustained in Friday night's 4-1 home win over Minnesota. Larsen's helmet was cracked when he was struck by a Minnesota slap shot. ... Lehtonen has started 16 of Dallas' last 18 games. ... C Jamie Benn (leg), LW Brenden Morrow (neck, shoulder) and C Jake Dowell (finger) are out for Dallas. ... Vancouver is without D Keith Ballard (concussion), LW Andrew Ebbett (collarbone) and LW Aaron Volpatti (shoulder). ... As Monday's deadline approaches, several Stars have been trade targets, including Morrow, Ribeiro and Ott. Vancouver has been mentioned among the teams that might be interested in Morrow and/or Ott.