Sundqvist, who entered the game with two goals in 72 NHL games, gave the Blues a 3-2 lead at 6:29 of the second period, beating Fleury above his shoulder from the left circle.
"I actually thought it was going to hit the (Golden Knights defenseman Deryk Engelland)," Sundqvist said. "I was happy that it went in."
Colton Parayko extended the lead to 4-2 at 14:49 with a one-timer from the left circle.
Tuch cut it to 4-3 at 8:02 of the third period with a power-play goal, but Bozak beat Fleury glove side from the left hash marks to make it 5-3 at 11:47.
"I think [Joel Edmundson] made a great play," Bozak said. "If it was a pass or shot, either way, he got it past the guy to the net towards me, so it was a great play by him. The goalie kind of went down thinking it was a shot, so I had quite a bit to shoot at."
Sundqvist gave the Blues a 1-0 lead at 2:02 of the first period, holding off Golden Knights defenseman Brayden McNabb and scoring into an open net. Pierre-Edouard Bellemare tied it 1-1 at 6:29.
Vladimir Tarasenko scored a power-play goal to make it 2-1 at 3:11 of the second period, but Erik Haula scored 23 seconds later to tie it 2-2.
Blues forward Robby Fabbri had an assist in his first NHL game since Feb. 4, 2017. He was medically cleared this offseason following his second ACL surgery.
"I felt like I wasn't nervous," Fabbri said. "I felt like I was able to play my game and I think those three games in San Antonio (of the American Hockey League) helped with my legs and get my legs underneath me and it felt good tonight.
"It was exciting. It was exciting for me and my family. They were all together watching tonight. It's just great to get back in the lineup with the guys. It makes it even more special. We got a big win tonight."
