Elliott turned back the Sharks despite being under pressure most of the night. The biggest of his 13 saves in the final period was stopping Dan Boyle's shot during a power play.
"The puck kind of squirted loose and he had a clear shot," Elliott said. "It was a desperation play to get over and get anything in front of it. It happened to go into my glove and I'm happy for that. Those are the fun saves to make."
Greiss, who played in San Jose's loss to Anaheim on Friday night, was back in the net although Antti Niemi said he felt good enough to play.
"The puck seemed to be bouncing off the ice pretty badly," Boyle said. "You have to keep it simple, especially on bad ice. We started to do that in the second period. We just need to simply things and keep playing hard."
D'Agostini picked off a pass in front of the Sharks net at 2:37 of the first period and beat Greiss to give the Blues the early edge.
Handzus tied the game with a power-play goal in the second period, picking up Ryane Clowe's rebound and getting it past Elliott.
Burns' shot on the net deflected off a defenseman and floated past Elliott to give the Sharks a 2-1 lead in the second period.
Huskins picked up a loose puck along the boards and fired an unassisted goal past Greiss into the upper left corner to tie the score at 4:33 of the third.
"I was just trying to get it to the net," Huskins said. "It just kind of found its way through."
A nifty give-and-go between Steen and Kevin Shattenkirk resulted in the go-ahead goal at 9:18.
Backes added an empty-net goal in the final minute.
Huskins scored his first goal against the Sharks in 16 career games. ... Clowe has three points in his first three games. ... Jim Vandermeer made his Sharks debut. ... Blues F Andy McDonald missed the game.
Copyright by STATS LLC and The Associated Press
Arena: SAP Center Location: San Jose, California
Referees: Dennis LaRue, Gord Dwyer Linesmen: Andy McElman, Bryan Pancich
Attendance: 17,562 (100.4% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Matt D'Agostini (2)
Michal Handzus (2)
(Power Play) Assists: Ryane Clowe, Jason Demers