Sherwood tied it 1-1 at 1:11 of the third with a wrist shot from the top of the right circle over Binnington’s glove.
Boeser gave Vancouver a 2-1 lead at 6:35 with his 200th NHL goal, a wrist shot from the edge of the right circle.
“He’s got a deadly shot,” Tocchet said of Boeser. “When he gets hot, look out.”
The Blues then scored twice in 24 seconds to take the lead.
Tucker tied it 2-2 at 9:28 with a snap shot from the high slot directly off a face-off win by Brayden Schenn, and Holloway tapped in a centering pass from Jordan Kyrou on the rush to take a 3-2 lead at 9:52.
“As soon as I touched the puck, 'Rouzy' was yelling for it, so I knew that he had some speed," Holloway said. "He usually gets excited like that when he's buzzing up the ice. So as soon as I passed it to him, I tried to get on my horse and go back post. He made a hell of a pass over and all I had to do was tap in."
But Boeser would get the Canucks a point when he was able to beat Binnington with a slap shot from the right circle and tie the game 3-3 with just four seconds remaining in regulation.
“It’s a playoff game,” Sherwood said. “(Great) resiliency and character to come back. We’ve still got to figure out our overtimes obviously, but we’re still looking for 60 minutes. When we put our foot on the gas, we showed what we can do like the other night against Winnipeg (6-2 win on Tuesday). It wasn’t enough tonight. But it was good character and grit to tie it up. We’ve got to be better for a full 60.”