Following a goalless first-leg draw in the Premiership/Championship play-off at Tannadice on Thursday night, the return at Paisley came to life midway through the first half.
United striker Nicky Clark broke the deadlock with a 23rd-minute penalty before Saints forward Danny Mullen – who became a father for the first time in midweek – levelled with a fine volley three minutes later.
With no additional goals, and with the home side reduced to 10 men in the 114th minute when substitute Duckens Nazon was given a straight red card by referee John Beaton for leading with his elbow in an aerial challenge with United defender Mark Connolly, the season for both teams boiled down to spot kicks.
Hladky made saves from Peter Pawlett, Pavol Safranko and Callum Booth, while Osman Sow hit a post.
Paul McGinn and Mihai Popescu scored for the home side, rendering Mateo Muzek's own strike against a post as irrelevant, allowing Oran Kearney's side to fight in the top flight for another season.
The Saints almost opened the scoring in the seventh minute when left-back Lee Hodson's low volley from 25 yards was pushed away by United keeper Benjamin Siegrist.
But it was the visitors who made the breakthrough after Hladky had made a fine save from Safranko's close-range drive.
Booth's attempt to return the loose ball from the byeline was stopped by the hand of Saints defender Jack Baird, resulting in Beaton pointing to the spot.
Clark stepped up to slot home but with the United fans still celebrating, Mullen took advantage of Connolly's sliced clearance to steer a volley low past Siegrist from 16 yards.
Striker Kyle McAllister later had the ball in the United net with a drive, only for the offside flag to go up.
Following a swift St Mirren break soon after the break, Mullen set up McAllister at the back post only for him to shoot wide.
Neilson then brought on Pawlett for Jamie Robson before Sow replaced goalscorer Clark.
In between, Hladky made a diving save from a Paul Watson header following a Paul McMullan corner.
At the other end, Siegrist made saves from Cody Cooke and Paul McGinn's powerful header.
In extra-time, there was more work for the United keeper who reacted quickly after a shot from Nazon, who had replaced McAllister, deflected off Connolly.
The second-half of extra-time saw Siegrist turn a powerful long-distance drive from Baird past the post for a corner before Nazon was sent off.
Then came the almost inevitable penalties with Hladky emerging the Buddies hero.