The defeat brings an end to the Dragons’ unbeaten record at Rodney Parade in the Celtic League this season and seriously harms their title chances.
Both sides failed to bring the contest to light in bitterly cold conditions that seemed to freeze any creative play.
The home side took the lead thanks to two penalties from Craig Warlow but they were unable to dominate the game even when Edinburgh prop Craig Smith was sent to the sin bin.
Edinburgh’s plight looked even more bleak when Simon Webster was forced to leave the field with a knee injury.
The Dragons began the second period with renewed vigour and soon notched the first try of the game.
The visitors were once again down to 14 men with skipper Todd Blackadder in the bin, when Dragons skipper Jason Forster finished off a barrage of attacks on the Edinburgh line. Warlow obliged with the conversion.
But they could not capitalise on the score and instead it was the Scots who dragged themselves back into the game. Replacement Rob Dewey led a breakaway raid for Laney to send full-back Derrick Lee over for a try, converted by Laney.
Edinburgh were soon pressing for the outright lead when number eight David Callam slipped through a gap in a wilting Dragons defence to cross for a try. The conversion by Laney gave Edinburgh a 14-13 lead moving into the last 10 minutes.
However, the Dragons edged their noses back in front when Warlow added a simple penalty after Laney had been penalised.
But in one final twist Laney stepped up to slot his last gasp match-winner in the eighth minute of added time.