The Dark Blues' substitute settled a nervy SPL affair when his free-kick from 30 yards deflected off a Hearts defender past keeper Jamie MacDonald.
Dundee improved after a slow start and their first opportunity also came from a free-kick, which MacDonald pushed over the bar from Kyle Benedictus after Andy Webster was booked for bringing down Carl Finnigan.
The hosts appealed for a penalty when Jamie 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 striker.
Moments later Webster caught John Baird from behind as he led with his arm in an aerial challenge and Beaton produced a second yellow for the Hearts skipper after he eventually picked himself up off the deck.
Dundee cranked up the pressure and Baird and Lewis Toshney fired wide after half-cleared corners before MacDonald saved Jim McAlister's header.
Conroy then fired narrowly wide from 25 yards and Declan Gallagher headed a decent chance over from 12 yards just before the decisive goal.
Dundee, whose win provides them with the narrowest of lifelines after Hibernian's 3-3 draw with struggling St Mirren, had sub Nicky Riley dismissed in the final minute for pushing Kevin McHattie.