Robert Bortuzzo and Alex Pietrangelo scored for Central Division-leading St. Louis, which killed four Chicago power plays as it rebounded from a 4-2 home loss to lowly New Jersey on Friday. Jaden Schwartz had two assists.
"I think tonight was a big game for us," Allen said. "We felt as a group the last few games weren't our best."
Added coach Craig Berube. "It was good to see. We were focused in practice. We weren't happy about losing to Jersey."
Allen was tested only in spurts in his 21st career shutout and his second this season over Chicago. The Blues swept their season series from the Blackhawks for the first time since St. Louis entered the NHL in 1967-68. For 2019-20, the Blues won won all four game and outscored Chicago 16-8.
"We were pretty solid defensively," Berube said. "We did a good job in our stand out at the blue line and not allowing entries and things like that. All the the things you have to do to keep them off the scoreboard and keep the shots down."
Corey Crawford had 24 saves in his ninth straight start as Chicago lost its second straight following a four-game winning streak. The Blackhawks' power play hasn't connected in its last 14 chances.
"We would have loved to get one on the power play," Chicago coach Jeremy Colliton said. "We had our opportunities and didn't finish. I thought (Crawford) was really good."
Both teams had a handful of prime chances in the scoreless first period.
Crawford sprawled to thwart Mackenzie MacEachern on an initial shot from the left circle then a rebound about 8 minutes left in the period
Bortuzzo gave the Blues a 1-0 lead at 6:41 of the second to cap a flurry around the Chicago net. It was just his second goal of the season and first in two months.
After Crawford stopped Schwartz's wraparound from the right side, the rebound trickled into the slot. Brayden Schenn swatted the puck to the left side, where Bortuzzo pitch-forked it over Crawford's shoulder from a sharp angle.
"I don't know what the geometry was on that one, but I'll take it," Bortuzzo said.
Dylan Strome's shot glanced off the crown of Allen's mask midway through the period. Allen went to the bench briefly, had his mask tweaked and returned to the crease.
Kane skated in alone on Allen with 13:50 left, but Allen came up with a nifty left pad save.
Oskar Sundqvist elbowed Blackhawks defenseman Adam Boqvist in the head along the boards with 5:01 left in the second to set up Chicago's fourth power play. It also led to fracas that fueled a free-swinging fight between Caggiula and Vince Dunn.
Boqvist didn't return after the hit and is under concussion protocol. Caggiula suffered a right hand injury in his altercation.
The Blues killed the advantage, then Allen stopped Alex DeBrincat point-blank in the final seconds of the period.
Pietranglo gave the Blues some breathing room when he made it 2-0 at 5:37 of the third. After Nylander turned over the puck in the Chicago zone, Pietrangelo swept a shot past Crawford from the left side after taking Schwartz's cross-ice feed.
Allen came through with several close in saves in the final minutes after Crawford was pulled for an extra attacker.
Sundqvist was closing in on empty-net goals with 90 seconds left, but DeBrincat skated from behind to hook his stick.
The game telecast on NBCSN had an all-female crew of about 30, marking the first time a pro hockey game was presented solely by women in the United States. It coincided with International Women's Day.
Kate Scott handled play-by-play, calling her first NHL game, with Olympic gold medalists AJ Mleczko and Kendall Coyne-Schofield providing analysis. Kathryn Tappen and three-time Canadian Olympic gold medalist Jen Botterill anchored coverage from the studio.
"It was fast, it was fun," Scott said. "It was a night I'm going to remember forever."
Allen and the Blues also blanked the Blackhawks at the United Center, 4-0, on Dec. 2, 2019. ...St. Louis D Justin Faulk missed his second game with a upper body and is out day-to-day. ... Boqvist has just returned to the lineup Sunday after missing Friday's game at Detroit (right wrist) and the Blackhawks re-assigned D Dennis Gilbert to Rockford of the AHL. ... Chicago D Lucas Carlsson sat out second straight game after being hit in the head with a puck in Thursday's win over Edmonton.
Blues: Host Florida on Monday.
Blackhawks: Host San Jose on Wednesday.