Kucherov also extended his point streak to nine games (eight goals, 10 assists).
Steven Stamkos and Brayden Point each had a goal and an assist, and Brandon Hagel had two assists for the Lightning (30-20-5), who have won three in a row and 11 of their past 14 games. Andrei Vasilevskiy made 28 saves.
"The effort is there, these aren't perfect games by any means," Stamkos said. "But when you get to this stretch of the year every point is huge. I thought tonight was probably the best in terms of the defensive zone structure, especially against their top line, which is really dangerous."
Bowen Byram scored twice, and Artturi Lehkonen had a goal and an assist for the Avalanche (33-18-4), who went 1-4-1 in their six-game road trip coming out of the All-Star break. Justus Annunen made 33 saves.
"Not to make excuses for ourselves, but I think we're tired," Byram said. "It will be nice to sleep in our own bed and recoup a little before we play [the Arizona Coyotes on Sunday]. We're looking forward to that."
Stamkos tied it 3-3 at 4:00 of the third period, scoring on the forehand to finish a breakaway set up by an outlet pass from Kucherov.