Mark Wilson cancelled out Tom Prydie's penalty to put the visitors ahead while Beli Adulla extended their lead.
Sonatane Takulua kicked his second penalty of the afternoon in the second half before conditions deteriorated.
The game ended after 68 minutes as the pitch deteriorated.
As the game ran for longer than 60 minutes, the result stands, meaning Newcastle seized victory without the need for a replay.
It was the third match to be played on the pitch in five days after Newport RFC beat Swansea 21-18 in the Welsh Rugby Union Challenge Cup on Wednesday and Newport County beat Hartlepool 3-1 in football's League Two on Saturday.
Heavy rain overnight and during the course of the match on Sunday meant the surface gradually got worse until the early conclusion.
The result means the Dragons are out of semi-final contention in Pool One and while it was Newcastle's first win in the competition, they cannot progress to the knock-outs from Pool Four.
Newport Gwent Dragons: Carl Meyer; Adam Hughes (capt), Tyler Morgan, Adam Warren, Tom Prydie, Dorian Jones, Sarel Pretorius; Thomas Davies, Rhys Buckley, Brok Harris, Matthew Screech, Rynard Landman, Ollie Griffiths, Nic Cudd, Harri Keddie
Replacements: Thomas Rhys Thomas, Sam Hobbs, Lloyd Fairbrother, James Thomas, Robson Blake, Charlie Davies, Angus O'Brien, Sam Beard
Newcastle Falcons: Alex Tait; Belisario Agulla, Chris Harris, Juan Pablo Socino, Vereniki Goneva , Joel Hodgson, Sonatane Takulua; Rob Vickers, Scott Lawson, Jon Welsh, Calum Green, Evan Olmstead, Sean Robinson, Will Welch (capt), Mark Wilson
Replacements: Kyle Cooper, Adam Brocklebank, Andrew Foster, Mouritz Botha, Callum Chick, Michael Young, Craig Willis, Dominic Waldouck
Newcastle Falcons sealed victory against Newport Gwent Dragons in the Anglo-Welsh Cup despite the game being abandoned because of poor playing conditions.