Conroy's deflected effort found the net with eight minutes left of a nervy performance against 10-man Hearts, which left the hosts eight points adrift with three games left in the Clydesdale Bank Premier League.
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.
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.
Dundee have now only lost once 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 hosts claimed for a penalty when Jamie Hamill caught Carl 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.
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.
Half-time substitute Conroy was quickly involved, curling in a corner that Brian Easton headed off the bar. Another new introduction, Hearts midfielder Dale Carrick, was also 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.
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 Jamie MacDonald motionless as the ball bounced into the corner of the net.
Leigh Griffiths' late equaliser for Hibs against St Mirren keeps Dundee's hopes alive for at least another week. They were also reduced to 10 men in the final minute when Nicky Riley was sent off for pushing Kevin McHattie.
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