Binnington responded after Blues coach Craig Berube chose not to pull him after the fluky third goal. He made 16 saves over the second and third periods and three in overtime.
"He's our guy, we know that," Berube said. "There was no reason to pull him other than the momentum swing maybe for our team, but I didn't want to take him out for that purpose. I didn't feel that that was the right move. I can't fault him on the goals, so I'm not going to pull him."
Ryan O'Reilly cut it to 3-1 when he scored from the slot at 3:15 of the second.
"I think we just kind of reset on the bench," O'Reilly said. "There were bad bounces and they kind of were just shooting everything, going to the net, getting chaos, got some goals early and put us on our heels. But it was a good thing we settled and took a deep breath, all of us, and started to chip away, started getting back, winning battles and just kind of getting on our toes a bit more and started to build it."
Brown, who was recalled from Springfield of the American Hockey League on Sunday, scored on a wrist shot from the left circle through a screen at 6:13 to make it 3-2.
Barbashev scored in his third straight game, a power-play goal at 1:11 of the third period, to tie it 3-3.
O'Reilly scored the only goal of the shootout in the second round. Binnington made saves against Perry and Ross Colton after Victor Hedman hit the post in the first round.
NOTES:The teams will play again in Tampa on Thursday. … The two goals in five seconds set a Lightning record. They scored twice in nine seconds three times, most recently against the New York Islanders on March 25, 2016. … Perry extended his point streak to five games (three goals, two assists). ... Cernak played 19:48 after missing eight games because of an upper-body injury. ... Forward Alex Killorn played his 662nd game with the Lightning, tying Pavel Kubina for fifth in their history. ... St. Louis has won five straight against Tampa Bay, dating to Dec. 12, 2017. ... Binnington is 5-0-0 in his NHL career against the Lightning. ... Blues forward Pavel Buchnevich had an assist to extend his point streak to four games (four goals, three assists). He has scored 14 points (seven goals, seven assists) in the past 12 games. ... Barbashev has four goals during his goal streak. ... The Blues were without forward Tyler Bozak, who was placed in COVID-19 protocol Tuesday. … Brown's father, Jeff Brown, played for the Blues from 1989-94 and scored 294 points (80 goals, 214 assists) in 329 regular-season games.