The Canucks had a 1-0 lead heading into the closing 20 minutes in a bid to win their third in a row.
Smith tied the game on a slap shot 3:28 into the final period and Spaling put Vancouver in front while Alex Burrows was serving a two-minute penalty for hooking.
Carter Hutton made 33 saves to collect the victory, giving up only a second-period goal to Chris Higgins on a deflection.
Roberto Luongo allowed the two Nashville goals on 25 shots.
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