Goals from Logan Ndenbe and Daniel Salloi lifted Kansas City to victory, Ndenbe notching his second career goal a week after grabbing his first in Kansas City's 4-1 triumph in the first game of the best-of-three series.
Reigning MLS Cup champions Los Angeles FC took another step in their bid for a title repeat, edging Vancouver Whitecaps 1-0 thanks to a penalty from Denis Bouanga to win their series 2-0.
Expansion club St. Louis had defied all expectations in their debut season, sealing the top seed with weeks to spare.
Kansas City, meanwhile, needed a win on the final day of the regular season to make it into the wildcard round, having amassed just three points over their first 10 games before turning things around.
"I couldn't be more proud to be a part of this team, a leader of this team," Kansas City captain Johnny Russell said.
"Just to be a part of this when the guys have done everything they have done to turn this season around, it's honestly up there with the most enjoyable things in my career."
Eager to press home their advantage in the best-of-three series, an aggressive Kansas City dominated possession in the first half, and Ndenbe made sure they were rewarded in stoppage time.
Salloi fed Alan Pulido from the left and Pulido found Ndenbe with plenty of space to fire a left-footed shot past Roman Burki to put the hosts up 1-0 at the half.
The Belgian left back's second goal in as many matches came after 48 games for Kansas City without one.
With a desperate St. Louis pressing in the second half, Salloi doubled Kansas City's lead in the 73rd minute against the run of play.
Russell, breaking free on the right, slipped a well-timed ball into the path of Gadi Kinda, who laid off a pass to Salloi.
St. Louis pulled one goal back with Celio Martins scoring in the 86th minute with a ball that soared over keeper Tim Melia.
But Kansas City held them off through more than 12 minutes of stoppage time necessitated by an apparent head injury to Dany Rosero.
Eighth-seeded Kansas City will now play either the Houston Dynamo or Real Salt Lake in the single-elimination Western Conference semi-finals.
Houston lead their series 1-0 heading into game two on Monday.
Bouanga strikes again
In Vancouver, a penalty conversion in the 24th minute from French-born Gabon international Bouanga was enough to give LAFC the victory they needed to advance after a 5-2 win over the Whitecaps in game one.
Whitecaps players had argued to no avail after referee Tim Ford pointed to the spot following Tristan Blackmon's challenge on Mario Gonzalez.
Bouanga, the league's top scorer in the regular season, thought he had scored again six minutes into stoppage time, but on review it was called back for offside.
Before the ruling was made, Vancouver coach Vanni Sartini was sent off, incensed that Ford had collided with Whitecaps midfielder Alessandro Schopf, the tangle helping LAFC launch what appeared to be a scoring run.
Sartini was still fuming as he departed, kicking a couple of signboards on the way out.
LAFC's Canadian goalkeeper Maxime Crepeau produced five saves, including a diving stop on a long-range blast by Ryan Gauld in the waning seconds of the match.
LAFC next face either FC Dallas or Seattle, who play a winner-take-all game three on Friday.
After surviving a late-season slump to reach the playoffs, LAFC are seeking to become the first team since the Los Angeles Galaxy in 2012 to win a second straight title.
