The Blues came out of the period tied 1-1 because Tomas Hertl scored for the Sharks 34 seconds after Backes' goal, and San Jose outshot them 24-12 and had a huge territorial edge the rest of the way.
But the Sharks, who entered the series 13-for-42 (30.9 percent) on the power play in the 2016 Stanley Cup Playoffs, couldn't get the equalizer after Blues center Paul Stastny was called for hooking Joe Pavelski at 8:44 of the third period.
St. Louis defensemen Alex Pietrangelo and Jay Bouwmeester were on the ice for 4:16 of San Jose's six minutes of power-play time.
"They were hot in the last series, they have a lot of big numbers on the power play," Pietrangelo said. "We wanted to limit their chances on the power play. We took a couple; be nice if we took none. We stepped up to the challenge like that, it was a big topic for us all week. We executed the way we wanted to."
The Sharks power play burned the Los Angeles Kings and Nashville Predators in the first two rounds, but the Blues were relishing the chance to shut it down, or at least slow it down.
"It's huge," St. Louis defenseman Kevin Shattenkirk said. "We're really excited about this opportunity. We have a tough opponent over there on the power play, and it's something (where) we're going to have to be on our toes for the entire series because they're lethal. Tonight was just a great job."
They'll get another opportunity against the Sharks power play in Game 2 at Scottrade Center on Tuesday (8 p.m. ET; NBCSN, CBC, TVA Sports).