Binnington made 31 saves for the Blues, whose shutout streak ended at 233:50, the second longest in their history (258:29, set in March 2016). Binnington has won nine straight starts, a Blues rookie record.
Zach Hyman and Auston Matthews scored, and Andersen made 38 saves for Toronto (36-19-4), which went 3-2-1 on a six-game road trip. The Maple Leafs trail the Boston Bruins by two points for second place in the Atlantic Division.
"I think we can be proud of our effort," Matthews said. "All in all, battling back there in the third to score two goals and force it into overtime and get a point. It could have gone either way, but we kind of figured it out halfway through the second and in the third and started playing a lot better."
St. Louis took a 2-0 lead for the seventh straight game when Schwartz (10:13) and Parayko (17:42) scored in the first period. The Blues outshot the Maple Leafs 19-9.
Parayko's goal, which came on the power play, needed a review after it wasn't initially called a goal on the ice.
"I thought it hit the post, because I didn't see anybody react and the goal horn go off or anything," Parayko said.