Saad scored in his fourth straight game to give the Blues a 4-3 lead at 2:04 of the third period when Jordan Kyrou found him in the low slot for the quick shot.
"It was just a good forecheck by [Brayden Schenn]," Saad said. "They had pressure on the puck and 'Rouzy' ended up with it. I just kind of found open ice and he made a great pass to me in the slot.
"They just seem to be going in. I think the belief and how we're playing confident as a group, obviously playing with some good players there. Getting the opportunities and it's nice to finish them."
Rasmus Andersson thought he had tied it 4-4 at 2:19 on a wraparound, but the Blues challenged, and the play was deemed to be offside after video review.
Kuzmenko also had a goal waved off 1:11 into the third that would have given the Flames a 4-3 lead on a high stick.
"We made a good push," Kuzmenko said. "The play gets blown dead. Could have played on and get reviewed at least to get the right call. We let them score right after so it's on us. We let momentum swing their way. We made some good pushes but not enough."
Zack Bolduc's second NHL goal gave the Blues a 1-0 lead at 13:02 of the first period on a backhand from the slot through Wolf's pads.
Kuzmenko tied it 1-1 at 14:48 after he stripped St. Louis defenseman Justin Faulk from behind and beat Binnington high on the short side.
Huberdeau made it 2-1 at 18:37 when he tipped in Weegar's point shot on the power play.
Neighbours tied it 2-2 at 3:27 of the second period after finishing off a Colton Parayko right point shot, a Robert Thomas tip in front to Neighbours on the rebound.
"Set face-off play," Neighbours said. "They kind do whatever they want, I go to the net. It was a good shot. 'Tommer' got a stick on it and it bounced right to me."