The home side came back strongly and - after a sustained assault on the Ospreys' line - Steve Jones peeled off a maul and fed a delightful pass to Lewis for the first try.
Biggar briefly restored the visitors' lead with a second penalty, before the Dragons piled into the lightweight pack selected by coach Sean Holley, man-of-the-match Joe Bearman to the fore.
Charteris drove through a ruck to get his side close to the try line, then Thomas picked up to power over.
The Wales prop lost possession under Ryan Jones's challenge, but the ball went backwards and Thomas was able to regather for the try.
The Dragons looked to be benefiting from the backroom input of ex-Ospreys coach Lyn Jones as they picked the visitors' set plays and drove them back heavily in the tackle, Shaun Connor extending the lead with a penalty.
Gavin Henson makes a try-scoring return to action for the Ospreys
The Rodney Parade crowd's confidence was jolted, though, as Tommy Bowe glided in from the wing onto Biggar's pass before drawing the cover and sending Henson cantering over for a try.
Biggar missed the conversion, but landed a 39th-minute penalty to cut the deficit to a point at half-time.
Connor landed his second penalty straight after the break, prompting the Ospreys to introduce Filo Tiatia and Mike Phillips from the bench, Wales scrum-half Phillips making his first appearance in nine months after his knee ligament injury.
But it was the Dragons who struck when the visitors hesitated as they wrongly thought there had been a knock on in their 22.
Hoani MacDonald quickly picked up and fed Wyatt, the wing celebrating becoming the first man to make 100 Magners League starts by scoring his 25th try in the competition.
The Ospreys fired straight back, Sonny Parker picking a great angle to power through the home defence before off-loading to Henson for the full-back's second try.
More than anything it was probably a moment of madness
Colin Charvis on his kick
Biggar converted, but blew a simple penalty that would have restored his team's lead.
He was then isolated trying to run ball from his own half, and from the turnover that followed veteran flanker Charvis produced a surprise, pin-point chip over the Ospreys defence.
It landed in Jason Tovey's waiting arms, the full-back strolling in for a bonus-point try despite the Ospreys' unheeded calls for offside.
The visitors won a penalty from the restart, Biggar converting to reduce the gap to six points and give his side hope of just a second win in six visits to Newport.
But the only consolation was a losing bonus point that lifted the Ospreys above Glasgow to second in the table.
Dragons flanker Colin Charvis:
"More than anything it [Charvis's kick] was probably a moment of madness, my left boot has caused quite a few problems for teams in the past.
Turner delighted with Dragons victory
"I just saw that Tovey was unmarked and free, I just thought kick it and it went into his hands.
"We've had a really difficult few weeks [against] European Cup winners and Magners League winners.
"It's just a case of we've been battling and battling and battling and trying our best to get a win and... it happened for us and it was a great effort by the team."
Ospreys coach Sean Holley:
"Very disappointing to lose a game when you know I think for large periods we were the much better team.
Holley disappointed as Ospreys crash
"Difficult conditions and some things didn't go our way tonight Phil [Steele], I don't know why, we'll have to have a little look at it but we've got a pretty disappointed dressing room.
"It was a great spectacle for the neutral, lots of thrills and spills out there, but we're in the winning business we are and that the only thing we can take out of this is that we got a point."
Dragons: Tovey, Wyatt, Gomer-Davies, Sidey, Fussell, Connor, R. Lewis, Black, S. Jones, R. Thomas, A. Jones, Charteris, Charvis, MacDonald, Bearman. Replacements: Smith for Sidey (68), Isaacs for R. Lewis (62), L. Evans for A. Jones (62), Lydiate for Bearman (74). Not Used: Willis, Robinson, Dolman.
Ospreys: Henson, Bowe, Parker, A. Bishop, Vaughton, Biggar, Wells, James, Shervington, Griffiths, Gough, Thomas, Smith, Holah, R. Jones. Replacements: Beck for A. Bishop (69), K. Phillips for Wells (50), A. Jones for Griffiths (52), Tiatia for Smith (47). Not Used: Breeze, Tandy, M. Phillips.
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