WATCH: [All Blues vs. Sharks highlights]
The Sharks are one point ahead of the Anaheim Ducks and two ahead of the Los Angeles Kings.
"I think our motivation was the standings," Sharks coach Peter DeBoer said. "Everyone's fighting for a playoff spot. We know how important points are, and the guys were ready to play.
"It wasn't an easy game. The Blues battled hard, their goalie made some big saves. What I liked was our resiliency. We stuck with it, didn't try and force it, knew that if we stayed with it long enough it would come, and eventually we got rewarded."
Boedker took a pass from Joonas Donskoi and beat Jake Allen with a wrist shot for a 1-0 lead at 12:12 of the third period. The goal was the Sharks' first on the power play in 13 games (1-for-29).
"We got set up, we got it to move around a little bit and [Donskoi] made a great play," Boedker said. "I got good wood on it, and it definitely felt good to get that in the back of the net.
"We knew about the opportunities that we'd missed. We know what's going on. It's just a matter of sticking with it. We have All-Star players in this group that are going to dig us out of the hole."