After a scoreless first two rounds of the tiebreaker, Raymond dented Sergei Bobrovsky when he pulled the goalie to the left, stopped, spun around and slipped the puck into the empty right side of the net.
Luongo then wrapped up the win with a nifty poke-check on Mark Letestu after stopping all 11 shots he faced in the third period and overtime.
Jason Garrison scored in regulation for Vancouver, which broke a four-game losing streak.
Letestu scored the second period and Bobrovsky finished with 37 saves as Columbus saw its five-game winning streak come to an end.
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