Dundee claimed for a penalty when Hamill caught Finnigan with a high boot in the face but referee John Beaton restarted with a drop ball after stopping play for treatment to the former Falkirk striker.
Dundee looked devoid of ideas and were lacking composure in the final third but Conroy sent the home fans into wild celebrations when his free kick appeared to take a deflection and left MacDonald motionless as the ball bounced into the corner of the net.
Conroy fired just wide from 25 yards before Hibs regained the lead and Declan Gallagher headed a decent chance over from 12 yards just before St Mirren levelled at Easter Road.
Half time substitute Conroy was quickly involved, curling in a corner that Brian Easton headed off the bar.
Live blog: Saturday Matchday Live April 27 Dundee have now only lost once Nike Air Max 90 Ultra Moire Triple White
Hearts, without the injured Danny Wilson, were also chasing a hat trick of victories and they had the first chance from a 20 yard free kick but Steven Simonsen held Jamie Hamill's powerful drive.
in eight SPL games since Brown took over from Barry Smith on a short term basis, a record that earned the former Dundee player a two year contract extension on the eve of the match.
The home physio was called into action again seconds later when Webster caught John Baird from behind as he led with his arm at an aerial challenge in the centre circle. Both players went down in agony but more misery awaited the Hearts defender as Beaton showed him a second yellow and red card when he got back on his feet.
Dundee improved after a slow start and their first opportunity also came from a free kick after Webster was booked for bringing down Carl Finnigan.
Conroy's deflected effort found the net with 10 minutes left of a nervy performance against 10 Nike Air Max 90 Infrared Ultra
Dark Blues live to fight another day with late win over ten man Hearts
The joy was short lived as the Hearts fans were soon cheering a third goal for St Mirren, but the Dundee fans were soon celebrating again though with news of another Griffiths goal to keep their hopes alive for at least another week.
Hearts manager Gary Locke had moved Darren Barr back to fill the space left by Webster and he made a change with Mehdi Taouil replacing striker Michael Ngoo, who then received treatment for his shoulder injury.
Kyle Benedictus powered a rising strike from 30 yards which Jamie MacDonald pushed over the bar.
DUNDEE averted relegation for another week after Ryan Conroy's 30 yard free kick earned them a third consecutive victory.
Another new introduction, Hearts midfielder Dale Carrick, was also Nike Air Max Green And Grey in the heart of the action when he ran on to Jason Holt's pass, but the angle proved too tight and he shot wide.
man Hearts, which left the hosts eight points adrift with three games left in the Clydesdale Bank Premier League.
Dundee, without the influential Gary Harkins, struggled to exploit their man advantage after Andy Webster was sent off in the 28th minute but substitute Conroy produced the goods as a rollercoaster affair at Easter Road kept the Dens Park crowd on their toes.
The Dundee fans were soon chanting the name of former Dens Park striker Leigh Griffiths after news broke that he had opened the scoring for Hibs. But they had little else to shout about for the remainder of the half and the Great Escape theme tune that greeted Dundee's re emergence for the second half did not appear to resonate much with the home fans in the 5,896 crowd.
But with St Mirren securing a 3 3 draw with Hibernian, Dundee need to win all of their remaining games and hope St Mirren lose all of theirs as the Paisley side are 24 goals better off than John Brown's men.
The home side cranked up the pressure and Baird and Lewis Toshney fired wide after half cleared corners before MacDonald saved Jim McAlister's header.
The Hearts fans were quick to reinforce Dundee's predicament after word emerged that Esmael Goncalves had levelled for St Mirren. Brown brought on Nicky Riley and Colin Nish as Dundee pressed for the winner they Nike Air Max Classic Mens
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