The altercation, that occured after a first half scrum collapsed, resulted in Glasgow skipper Jon Petrie being taken to hospital with a shoulder injury and the Dragons' Rhys Thomas receiving a yellow card.
In a bruising opening period Glasgow's Dan Parks had two opportunities to put his side ahead but faile don both occasions.
To add to the Warriors' woes Canadian international prop Kevin Tkachuk was forced off with a shoulder problem.
The mass brawl appeared to spark some creativity with first Glasgow notching a try through flanker Steve Swindall. Parks was back on target with the simple conversion.
However, Glasgow were unable to build on their try and the Dragons dragged themselves back into the game after the break.
Dragons skipper Jason Forster grabbed a try from close range and Craig Warlow added the extras from a difficult angle.
The vistors then took the lead with another try from Gareth Wyatt and again Warlow was on target with the conversion.
Parks pulled three points back with a penalty but the Dragons reposnded with another try from Richard Bryan after he pounced in a loose ball.
Glasgow rallied again with a five-pointer from debutant Mike Roberts but the game was lost.
Glasgow's Jon Petrie gets to grips with the Dragons Rhys Thomas