Halischuk raced down the right side after the Predators won a neutral zone face-off, collected a pass from Nick Spalling and fired a wrist shot from the circle that beat Roberto Luongo 14:51 into the second extra period.
Ryan Suter forced the overtimes with only 1:07 remaining in regulation when his soft shot from the corner dribbled past Luongo as Nashville rebounded from a 1-0 loss in Game 1.
Pekka Rinne was spectacular in net for Nashville, making several acrobatic saves in overtime while turning aside 32-of-33 shots.
Alexandre Burrows got the Canucks' only goal 2 minutes into the contest.
Luongo finished with 44 saves for Vancouver, which will try to regain the momentum when the series switches to Nashville for Tuesday's Game 3.