Patrick Kane scored, and Alex Stalock made 26 saves for the Blackhawks (8-22-4), who are 1-10-0 in their past 11 games and 2-17-1 in the past 20.
"We've got to keep working at it," Chicago coach Luke Richardson said. "The work ethic was great tonight. Keeping them to four or five shots in the third period and still generating a lot in the other end is a good sign. But the guys have to keep their heads up and not get frustrated, otherwise it'll get harder. We've got to push through and one of these days we'll get a good bounce and it leads to a win."
Leivo put the Blues ahead 1-0 at 3:12 of the first period when he converted from an open slot after Blackhawks defenseman Jack Johnson's reverse attempt to move the puck from the corner hit referee Marc Joannette.
St. Louis' fourth line of Alexey Toropchenko, Nathan Walker and Leivo created the good fortune with its forecheck.
"I'm a big believer in that, and you pretty much hit the nail on the head," Walker said. "I just kind of scooped up the puck and I would have had to turn my hips to try and get a shot off. So, I'm standing right there and figured it'd be a good play to hand the puck over to [Leivo]."
Kane's power-play goal at 9:13 tied it 1-1 when he was able to skate into the left face-off circle and beat Binnington with a wrist shot through traffic.
"Special teams were pretty good. For the most part, I thought we did a decent job," Blackhawks defenseman Seth Jones said. "[But] we're not scoring 5-on-5 a lot. It's on us. We're only putting one or two in in a game. Mistakes are more magnified when we don't put the puck in the net."