Matt Duchene had a goal and two assists for Colorado (12-2-0), which fell one win short of making NHL history.
The Avalanche were attempting to set a franchise record and match the 2007-08 Ottawa Senators for the best 14-game start in league history.
The Predators appeared to take control when Gaustad and Wilson scored early in the second to give them a 4-2 lead. Paul Stastny got one back with a power-play goal with 6:27 left in the period, and Duchene tied it 4:23 into the third.
Wilson answered with his second goal of the night when he beat Semyon Varlamov at 7:34.
Varlamov finished with 22 saves while allowing a season high in goals.
The Avalanche pulled Varlamov for an extra skater with more than a minute left, and Gabriel Bourque scored into the empty net.
Nashville goalie Carter Hutton made 11 of his 26 saves in the third to preserve the win.
The teams, now in the same division, found a reason to ramp up the rivalry when Colorado forward Jamie McGinn left his skates to check Nashville rookie defenceman Seth Jones into the boards. Jones stayed on the ice for a moment before skating to the bench under his own power.
He went to the locker room but returned late in the period.
McGinn was given a charging minor and the Predators capitalized with Smith's third goal to give Nashville a 2-1 lead.
Stalberg opened the scoring with his first goal of the season at 13:12 of the first, and P.A. Parenteau tied it with his fifth 2:11 later.
NOTES: Ryan Ellis and David Legwand had two assists each for Nashville. ... McGinn returned to the lineup after missing three games with a knee injury. ... Nashville C Matt Hendricks missed his second game with an upper-body injury while C Mike Fisher missed his first game with a lower-body injury. Both players are day to day. ... Avalanche LW Alex Tanguay will miss two weeks with a knee injury sustained in Saturday's win over Montreal.
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