O'Reilly scored his 11th goal of the season at 12:00 of the third period on a backhand from the slot.
The Blues had a continuous cycle going in the Coyotes zone, and Brayden Schenn slotted a pass to O'Reilly. Raanta got a piece of O'Reilly's backhand, but it caromed off the post, back off Raanta's arm and into the net.
"It's kind of weird with all the shots we threw at them that that was the one that went in," O'Reilly said. "Kind of a lucky bounce. I just tried to elevate it and it rolled over his shoulder, hit the post, hit his back and then in. It was kind of the bounce we needed. We did a lot of things well tonight against a tired team."
The Blues outshot the Coyotes 19-4 in the second period and had a power-play goal by David Perron overturned at 9:55 when video review determined Perron was offside.
St. Louis outshot Arizona 18-2 in the third period and 37-6 over the final 40 minutes.
"Obviously it was very tough to generate any offense tonight," Coyotes forward Christian Dvorak said. "Raanta was very good for us. We could have gotten run out of the building, so it was a great game by him. We just couldn't get anything going."