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.
Celebrity Families of 2014 [PHOTOS]