Mikko Rantanen had a goal and an assist, Colin Wilson had three assists, and Pavel Francouz made 44 saves for Colorado (7-0-1), which is the only team in the NHL without a regulation loss.
"It feels awesome. It's pretty special," Jost said. "It's kind of something you always dream about as a kid playing in your backyard. It felt really special. It was an awesome moment."
WATCH: [All Avalanche vs. Lightning highlights]
Yanni Gourde and Victor Hedman scored, and Andrei Vasilevskiy made 18 saves for Tampa Bay (4-3-1).
Jost made it 1-0 at 3:09 of the first period with a one-timer off a pass from Samuel Girard.
Gourde tied the game 1-1 at 17:35 with a redirection of Ryan McDonagh's shot.
Jost gave the Avalanche a 2-1 lead at 3:17 of the second period, scoring short side on Vasilevskiy from the left face-off circle.
"In this league, you've got to play a full 60 minutes, and tonight we took a couple of minutes off, and they capitalized on their chances," Lightning center Anthony Cirelli said. "Obviously, we got a lot of shots, 40-plus or whatever, and their goalie made some saves."
Gabriel Landeskog made it 3-1 at 5:48 of the second with a redirection of Nathan MacKinnon's shot, and Jost completed the hat trick 18 seconds later to extend the lead to 4-1 at 6:06.