The Avalanche outscored the Blues 20-7 and trailed for only 7:12 over the four games. Grubauer stopped 103 of 110 shots over the four games.
"I thought we got a little better as the series went along," Colorado coach Jared Bednar said. "It was a full 60-minute effort. The first three games we had some lapses in our game, some things that I didn't like. Tonight, I liked a lot."
Vladimir Tarasenko scored twice for St. Louis, which was swept for the first time since losing to the Los Angeles Kings in the second round in 2012.
Landeskog tipped in a shot from Samuel Girard at 14:53 of the second period to break a 1-1 tie.