Raymond tied it 4-4 just 48 seconds into the third period, scoring five-hole with a one-timer from the slot off a bumper pass from Veleno on a power play.
"That gave us momentum right away and something to build off," Raymond said. "We trust what we’re doing and it paid off."
Robby Fabbri gave the Red Wings a 5-4 lead at 6:38 when he redirected in Raymond's centering pass from the edge of the crease.
"I knew [Fabbri] was there," Blues defenseman Torey Krug said. "They flung a puck. I don't know how fast the puck was going, you guys have all the data, you can look it up. I didn't see it. It went in. Most times not. I hope that that doesn't go in and I can get a piece of it off my skate or something, but that's just what happened."
Rasmussen scored a short-handed goal into an empty net at 18:21 for the 6-4 final.
"Resilience in this room is something we pride ourselves in," Fabbri said. "We use that to our advantage. A win like this is a big one for the group.
"We’ve had a lot of guys step up in bigger roles. Games like this will take all of us, and games like this one feel the best."
Brandon Saad gave the Blues a 1-0 lead at 5:56 of the first period. He beat Husso, who was off his angle, from below the right circle off a cross-slot pass from Pavel Buchnevich.
Jake Walman tied it 1-1 at 13:33, roofing a rebound from a sharp angle over Binnington's left shoulder.
Jonatan Berggren gave the Red Wings a 2-1 lead at 15:31 on a rebound after Jeff Petry's initial shot hit the left post.