The game was given the go-ahead after several pitch inspections, and Justin Tipuric's fine try put Ospreys 7-0 up at half-time.
The mud-caked conditions hindered both sides but it was the visitors who adapted best and made fewest mistakes.
Dan Biggar's late penalty sealed the victory for the visitors.
As referee Ian Davies conducted his pre-match pitch inspection, Ospreys players and coaches seemed unhappy with the prospect of playing on such a sodden surface - on which football team Newport County had played less than 24 hours earlier.
But having won their last six matches in all competitions, the visitors entered this fixture with confidence and settled into an early groove with their forwards carrying diligently and full-back Sam Davies dominating the kicking exchanges.
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It was a loose Dragons kick which led to the opening score, as Tom Habberfield sparked an Ospreys counter-attack and, via an offload from the impressive blindside flanker Olly Cracknell, freed openside Tipuric to sprint clear for the first try.
Cracknell and the rest of the Ospreys pack seemed to relish the old-fashioned tussle in the mud, keeping things simple with ball in hand and restricting Dragons to just one shot at goal - a missed Angus O'Brien penalty - during a tight first half.
The second followed in the same vein, with the visitors' pragmatic and well-executed game-plan in contrast to that of the increasingly error-strewn Dragons.
The hosts were guilty of kicking the ball straight into touch on a number of occasions, and their inability to keep possession saw them pinned back in their own half for long periods.
Another Dragons mistake - Ollie Griffiths entering a ruck from the side - then presented Biggar with a straightforward penalty to seal the Ospreys' victory and deny their opponents a losing bonus point.
The Ospreys have now won 28 of the 30 Welsh derbies they have participated in.
Ospreys head coach Steve Tandy told BBC Wales Sport: "We're satisfied to get the win. Conditions were horrific.
"We found a way to win. It was only 10-0 but I think it was relatively comfortable."
Dragons head coach Kingsley Jones told BBC Wales Sport: "It's a disappointed changing room. The frustration is we did well in many aspects of the game.
"I thought in the second half the Ospreys kicked a lot but they came here to do a job. It was a clinical job by them and unfortunately we weren't quite as clinical.
"The difference in winning and losing high-level games is forcing errors but unfortunately we made four or five too many unforced errors."
Dragons: Carl Meyer, Pat Howard, Tyler Morgan, Sam Beard, Ashton Hewitt, Angus O'Brien, Tavis Knoyle; Phil Price, Elliot Dee, Brok Harris, Nick Crosswell, Cory Hill, Lewis Evans (c), Ollie Griffiths, Ed Jackson.
Replacements: Rhys Buckley, Sam Hobbs, Lloyd Fairbrother, Matthew Screech, Rynard Landman, Sarel Pretorius, Dorian Jones, Adam Warren.
Ospreys: Sam Davies, Hanno Dirksen, Kieron Fonotia, Josh Matavesi, Dan Evans, Dan Biggar, Tom Habberfield, Paul James, Scott Baldwin, Ma'afu Fia, Lloyd Ashley, Alun Wyn Jones (c), Olly Cracknell, Justin Tipuric, James King.
Replacements: Sam Parry, Nicky Smith, Rhodri Jones, Rory Thornton, Rob McCusker, Sam Underhill, Brendon Leonard, Jonathan Spratt.
Referee: Ian Davies (WRU).
Assistant referees: Simon Rees (WRU), Gwyn Morris (WRU).
TMO: Neil Hennessy (WRU).
Ospreys ground out a gritty win over Newport Gwent Dragons in a Pro12 derby played on a dreadful Rodney Parade pitch.