Stastny's 10th of the season 1:24 into the third got the Avalanche within one but Ryan Malone picked up his seventh goal for Tampa Bay and helped set up Sean Bergenheim to make it 6-3 with 4:05 left.
Colorado pulled within one again on goals by Yip at the 17:49 mark and Hejduk barely a minute later, but the rally stalled. Both teams missed scoring opportunities with the Colorado net unmanned.
"That was a big letdown," Boucher said. "We score some goals and especially when we're two or three goals ahead, we have a tendency to change things. We want to mature as a team."
Tampa Bay, which lost 8-1 to the Bruins on Thursday night, had dropped six consecutive games against Colorado.
"We were resilient right up until the end of the game," Avalanche coach Joe Sacco said. "Overall, for 60 minutes, we need to play better. There is no question about that. That is not good enough against that team tonight."
The Lightning got off to a fast start behind Nate Thompson, who beat goalkeeper Peter Budaj for the first goal of the game at 7:14 in the first.
Philippe Dupuis responded for Colorado, tying it at 1 with 4:07 left in the first.
Budaj finished with 33 saves. Dan Ellis had 19 stops for Tampa Bay.
Colorado managed only five shots in the first period, failing to get one on net until left wing Greg Mauldin attempted a wraparound with 8:37 to go.
Lightning D Victor Hedman reached 100 career NHL games. St. Louis is one shy of 800. ... Avalanche RW Milan Hejduk had two assists, extending his points streak to seven games. ... Scratches included LW David Koci and D Ryan Wilson for Colorado, and RW Steve Downie and D Matt Smaby for Tampa Bay. ... The Lightning improved to 7-2-2 at home. ... Tampa Bay is 11-0-0 on its annual Hockey Fights Cancer Night.
Copyright by STATS LLC and The Associated Press
Arena: Amalie Arena Location: Tampa, Florida
Referees: Wes McCauley, Chris Ciamaga Linesmen: Darren Gibbs, Jonny Murray
Attendance: 18,212 (94.8% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Nate Thompson (4)
Assists: Adam Hall, Victor Hedman
Philippe Dupuis (3)
Assist: Matt Hunwick
Martin St. Louis (11)
Assists: Simon Gagne, Mattias Ohlund
Mattias Ritola (2)
Assists: Johan Harju, Dana Tyrell
Matt Duchene (8)
Assists: Tomas Fleischmann, John-Michael Liles
Simon Gagne (2)
(Power Play) Assists: Martin St. Louis, Teddy Purcell
Paul Stastny (10)
Assists: Kevin Shattenkirk, Milan Hejduk
Ryan Malone (7)
(Power Play) Assists: Simon Gagne, Martin St. Louis
Sean Bergenheim (5)
Assists: Mike Lundin, Ryan Malone
Brandon Yip (6)
(Power Play) Assists: Daniel Winnik, Matt Duchene
Milan Hejduk (10)
Assists: Tomas Fleischmann, Kevin Shattenkirk