Dragons down holders Gloucester
Replacement scrumhalf Charlie Davies grabbed the decisive score to overturn a 21-18 deficit to add to five penalties from flyhalf Dorian Jones and one from fullback Carl Meyer.
Ben Morgan and Stevie McColl scored tries for Gloucester, with skipper Greig Laidlaw kicking three penalties and a conversion.
Victory for the Dragons gave them a semifinal spot against French Top 14 side Montpellier.
"By winning away, we've got the monkey off our backs," said Newport coach Kingsley Jones after the Welsh region picked up only their second away of the season.
Dragons were rewarded for their positive approach when they chose to go for the drive from a line-out instead of kicking a penalty to draw level.
"It was a big call from our skipper [Lewis Evans] but we came here to win and we didn't want to settle for a draw," said try-scorer Davies.
"At first they defended the line-out well, even though we had put all our backs in. I then heard a call to go to the blind side and I used my size to get over.
"No-one backed us to win here but I'm elated to score the winning try and I'm very happy for the team as a whole."
On Friday, Montpellier reached their first European semifinal when they defeated hosts Sale 25-19.
The reliable boot of Benoit Paillaugue proved crucial for Montpellier as he kicked five second half penalties to help turn round an 11-10 half-time deficit.
Former England flyhalf Danny Cipriani kicked two first half penalties to add to winger Tom Brady's try.
Montpellier, on a run of six successive wins in the French Top 14, had also grabbed a first period try through Paul Willemse.
But Paillaugue's deadly boot was enough for Montpellier despite a late try for Sam James.
The scorers:
For Gloucester:
Tries: Morgan, McColl
Con: Laidlaw
Pens: Laidlaw 3
For Dragons:
Try: Davies
Pens: D Jones 5, Meyer
Teams:
Gloucester: 15 Tom Marshall, 14 Rob Cook, 13 Henry Trinder, 12 Mark Atkinson, 11 Steve McColl, 10 James Hook, 9 Greig Laidlaw (captain), 8 Ben Morgan, 7 Matt Kvesic, 6 Sione Kalamafoni, 5 Mariano Galarza, 4 Tom Savage, 3 John Afoa, 2 Richard Hibbard, 1 Paddy McAllister.
Replacements: 16 Darren Dawidiuk, 17 Yann Thomas, 18 Paul Doran-Jones, 19 Elliott Stooke, 20 Gareth Evans, 21 Callum Braley, 22 Ollie Thorley, 23 Jacob Rowan.
Dragons: 15 Carl Meyer, 14 Adam Hughes, 13 Tyler Morgan, 12 Adam Warren, 11 Hallam Amos, 10 Dorian Jones, 9 Sarel Pretorius, 8 Taulupe Faletau, 7 Nic Cudd, 6 Lewis Evans (captain), 5 Nick Crosswell, 4 Rynard Landman, 3 Brok Harris, 2 Elliot Dee, 1 Phil Price.
Replacements: 16 Rhys Buckley, 17 Boris Stankovich, 18 Shaun Knight, 19 Matthew Screech, 20 Ed Jackson, 21 Charlie Davies, 22 Angus O'Brien, 23 Geraint Rhys Jones.
Referee: Pascal Gauzère (France)
Assistant Referees: Dudley Phillips (Ireland), Laurent Cardona (France)
TMO: Philippe Bonhoure (France)
Agence France-Presse