Duncan Keith had a goal and three assists, and the Blackhawks used a four-goal third period to finish off the Blues with a 5-1 victory in Game 6 on Sunday.
"It feels good to contribute in a big game," Keith said, "and I think just the main thing for me is just being able to move on and get a couple days to relax and just regroup and focus on the next round."
Chicago won four in a row after a slow start in St. Louis. The defending Stanley Cup champions will play the winner of the Minnesota-Colorado series in the Western Conference semifinals. The Avalanche lead the Wild 3-2 heading into Game 6 in Minnesota on Monday night.
Toews, Sharp and Shaw scored in the first 7 minutes of the third and Keith closed out the scoring as the Blackhawks improved to 14-2 in home playoff games over the past two seasons.
"They were dominating the first 40 minutes here and we came back with maybe the best period of the year," coach Joel Quenneville said.
T.J. Oshie scored for the Blues, who outshot the Blackhawks 36-27. Ryan Miller finished with 22 saves.
St. Louis went 0 for 6 in 10 minutes of power-play time over the first two periods, wasting a chance to take the lead. The Blues went 2 for 29 with the man advantage for the series.
"I think both the PK and (Crawford) won the game and the series ultimately," Quenneville said. "I think that was the big factor in us getting through."
The Blackhawks also struggled on the power play, but they scored when it mattered most.
With Jay Bouwmeester in the box for tripping, Keith made a nice stop to keep the puck in the St. Louis zone, and then sent a pass over to Toews. The captain beat Miller over his right shoulder for a 2-1 lead just 44 seconds into the third.
It was Toews' third game-winning goal of the series. He also scored on a breakaway in overtime of Friday night's 3-2 win.
Toews' 23rd career postseason goal seemed to take the air out of the Blues, and it got even worse for St. Louis. Sharp got loose for a breakaway, shook off a stick to the face by defenseman Kevin Shattenkirk and slid a shot past Miller.
"The third goal was really a backbreaker for us," Blues coach Ken Hitchcock said. "That was really the one that hurt. We had been chasing them all series and able to catch up in games, but I thought the third goal really took the wind out of our sails."
Sharp's first point of the playoffs sent a charge through the towel-waving sellout crowd of 22,144, and there were mocking chants of "Mil-ler! Mil-ler!" as Shaw added his second goal of the series and helped set up Keith for his second.
It was an eerily similar playoff exit for St. Louis to a year ago, when the Blues also were eliminated by the defending Stanley Cup champions in six games in the first round. In that 2013 playoff series, St. Louis won the first two games at home against Los Angeles, and then lost four in a row.
This year was supposed to be different, especially after the Blues acquired Miller from Buffalo on March 1. But they lost their last six games of the regular season, putting them in a first-round series against rival Chicago.
St. Louis rebounded for two 4-3 overtime victories, but the Blackhawks found their stride when the series shifted to Chicago. Crawford had a shutout in Game 3, Patrick Kane scored in overtime in Game 4, and Toews' breakaway score in St. Louis put the Blackhawks in position to advance.
"It was one play here, one play there," Blues captain David Backes said. "But it was on our stick and we needed to get the job done. We didn't get it done and we're going home too early."
Chicago defenseman Brent Seabrook returned from a three-game suspension. Seabrook was punished by the NHL for his hit on Backes in Game 2.
"It was tough. I can't watch hockey," Seabrook said. "You're hanging on every shot, every save, every play. You want to be out there helping the guys. It was obviously tough to watch it, but they did a great job."
Backes exacted a measure of revenge when he delivered a hard hit on Seabrook into the end boards in the second period. But Seabrook added two more assists and had six points for the series.
Blackhawks F Kris Versteeg was scratched after he played in the first five games. ... Keith, a top candidate for the Norris Trophy awarded to the NHL's top defenseman, tied a playoff career high with four points. ... Miller, who is eligible for unrestricted free agency, on his immediate future: "I don't know. I've just got to take things as they come right now. I guess I'm free to go to my sister-in-law's wedding. That's about it."