Stastny got the Blues going with a power-play goal early in the first period, his first score in 11 career games against Colorado. He spent eight seasons with the Avalanche before leaving for St. Louis.
Allen was shaken up with just more than three minutes remaining when Gabriel Landeskog's wrist shot appeared to hit him awkwardly. He was checked out by trainers before finishing off his third shutout of the season.
The last-place Avalanche were blanked for the 11th time this season.
Berglund scored on a power play in the second to give the Blues a 3-0 lead. It ended a seven-game goal drought and also marked his first since signing a five-year extension worth $19.25 million on Feb. 25.
Allen made one big save after another, especially in the second when he thwarted Rene Bourque's breakaway attempt by stopping the shot with his chest. With around 30 seconds left, he stopped another shot by Bourque.
This was the second half of a back-to-back for the Avalanche. They were routed 6-1 in Winnipeg, leading Landeskog to say the squad looked "like a junior C team."
Once again, the Avalanche came out sluggish in dropping their fourth straight game.
Stastny and Brodziak scored 1:37 apart in the first to help the slumping Blues get back on track. Stastny's goal was set up off a precision pass by Jaden Schwartz from down low. Stastny celebrated by pumping both his gloves. He still ranks ninth on the Avalanche's all-time points list (458).
Brodziak added another goal by firing in a shot after an Avalanche turnover near the net.
Forward Sven Andrighetto made his first appearance with the Avalanche. He was acquired in a deal with Montreal, but didn't join the team in Winnipeg as he waited for a visa.
He was thrown "right in the fire" -- coach Jared Bednar's description -- by being paired on the second line with Matt Duchene and Mikko Rantanen. Andrighetto turns 24 on March 21 and will be a restricted free agent at the end of the season.