It was Binnington's third straight win of the series after he replaced Ville Husso, who started Games 1-3.
"We had a good first period and kind of weathered the storm," Binnington said. "They came out hard and they were playing desperate. I think we matched that, and we came out in the second period and played really hard.
"It's a hard-fought series against a team that comes at you pretty hard, and we did a great job. We had guys step up, just a great team effort and it's good to be at this point. Now we're looking forward to the next challenge."
Cam Talbot made 22 saves in his first start of the series after Marc-Andre Fleury started the first five games. Minnesota, eliminated in the first round for the fifth time in the past seven seasons, was the No. 2 seed in the Central.
"It stinks], right," Wild coach Dean Evason said. "They loved playing together. There's no individuals, there's no [complaining] on the bench, there's no, 'I should get this, I should be on the power play.' They played for each other and that's what's so disappointing. It's disappointing we didn't win tonight and we didn't win in the first round of the playoffs. When you're that close, you're like, 'What's going to happen now? Are we all going to be together or not?' It's tough."
Talbot went 13-0-3 in his last 16 regular-season appearances.
"I was doing everything I could to stay ready," Talbot said. "You don't want to come in in an elimination game, but you still want the opportunity to play during the playoffs. They gave me the opportunity. Just wasn't good enough."
***[RELATED: [Complete Wild vs. Blues series coverage]*
Nick Leddy gave the Blues a 1-0 lead at 14:59 of the first period when he took a puck from the top of his own defensive zone circle, skated across the middle of the Wild blue line and fired a wrist shot past Talbot on the short side. Talbot appeared to be screened by defenseman Dmitry Kulikov on the play.
"The first goal was obviously big," St. Louis forward David Perron said. "It kind of took some pressure off there. It allowed us to go out and kind of realized what we had to do better and play better in the second period. We had maybe a few too many turnovers in the first."
The Wild said they felt they deserved a better fate than trailing after the first period.
"It was exactly what we were looking for; the first period was fantastic," Evason said. "We did everything that we could ask. We played right, played hard, got chances, just didn't score. They got that goal and we just uncharacteristically didn't handle it very well. Likely the difference obviously in the game."
The Blues outshot the Wild 21-5 in the second period when they scored three times.
Ryan O'Reilly's power-play goal at 9:26 made it 2-0, a shot from the slot after a pass from Brayden Schenn at the goal line.