Australian full-back Matt Burke made a dramatic return from injury to propel Newcastle through to their first Heineken Cup quarter-final.
The ex-Wallaby, who passed a late fitness test on his injured knee, kicked four goals and scored the match-winning try as the Falcons ended all Welsh interest in Europe.
Edinburgh provided the Dragons with a favour with their surprise win over Perpignan, but Chris Anderson's men could not help themselves thanks to an error-strewn display.
Falcons will now travel to Paris to face Stade-Francais in the last eight.
Burke revealed: ''Rob gave me up to kick-off to see how it was.
''It wasn't until the warm-up that I felt confident enough to go out there and play.
''There was no risk of damaging the knee any more. There were a couple of times when it felt a little bit sore but we've now got a week off and we'll do all the right stuff to get it right for a couple of weeks' time.
''We needed to win and we were running a little bit short on personnel but, if it wasn't right, I wouldn't have played. I felt I was reasonably up to it and luckily I've come through without a knock.''
Newcastle also came up with a patchy performance but had enough creativity to win with something to spare and will be encouraged by the likely return of Jonny Wilkinson for their tough away tie in April.
They were boosted by the late inclusion of both Burke and centre Jamie Noon, while Mark Wilkinson proved a useful fly-half replacement for his injured brother.
On the down side, they lost scrum-half Hall Charlton with a back problem after only seven minutes - while the Dragons were without captain Jason Forster through injury from midway through the first half.
Newport's loss was clearly the bigger blow, with James Grindal profiting from the space created by a workmanlike pack and relishing his duel with Gareth Baber.
The home side had first use of a stiff breeze and an abundance of possession, but a 13-5 interval lead barely looked enough.
Burke put his side ahead with a 15th-minute penalty but was wide with two more difficult attempts, while Tom May put a drop-goal attempt wide as the Falcons failed to turn territorial advantage into points.
Flanker Colin Charvis scored Newcastle's only first-half try, courtesy of a break by winger Michael Stephenson after the position had been set up by teenage centre Mathew Tait.
The visitors were down to 14 men at the time, with Rhys Oakley still waiting to replace the injured Forster - and their defence, not for the first time, was in total disarray as the Falcons moved the ball the full width of the pitch.
Burke added the conversion and stretched his side's lead with a second penalty in first-half injury-time after Newport centre Sione Tuipulotu was punished - and sin-binned - for cynically straying offside to prevent the Falcons making use of a clear overlap.
Burke, showing little sign of his knee problem, was almost a first-half tryscorer too. But his midfield counter-attack was denied by an ankle tap from Hal Luscombe.
Newport, who needed four tries to have a chance of securing a quarter-final spot, breached the Newcastle line just once in the first half - and even then it needed video official Bertie Smith to rule that winger Gareth Wyatt touched down before brushing the corner flag.
The Dragons managed to keep their line intact while down to 14 men early in the second half but cracked immediately upon Tuipulotu's return when sweet passing along the line engineered an overlap for May, who sped over at the corner.
As news began to filter through of Perpignan's impending defeat, Newport suddenly upped a gear - and Wyatt came off the right wing at a scrum to make the extra man as left-wing Kevin Morgan finished superbly.
Ceri Sweeney kicked the conversion to cut the deficit to six points, but that was as close as the Welshmen got. A break by Noon almost created another try for the Falcons before Burke's dummy opened up a yawning gap on 77 minutes and he strode majestically through for the all-important score.
He added the conversion to complete a personal haul of 15 points to render lock Ian Gough's late try mere consolation.
Rob Andrew paid tribute to Jonny Wilkinson's brother Mark, who recovered from a calf injury to don for the first time this season the number 10 jersey made famous by his younger brother and come up with an accomplished display.
''For Mark to step at 10 in a game of this intensity and to play as well as he did was just outstanding,'' said Andrew.
''He's got a lot of good skills. Pace-wise, perhaps he isn't in the same league as some of our other backs but he understands how we play and he's got good hands and good feet.
''All we asked him to do today was try and allow the rest of the team to function and I thought he did that brilliantly. He hasn't played in the first team this season but we felt he was our best option.''
The exit of Newport leaves Wales without a representative in the last eight for the first time, and coach Chris Anderson blames the lack of intensity in the Celtic League for his side's failure to make a mark in Europe.
''We gave 100 per cent but, because we don't compete at a high level week in, week out, it's tough to lift yourself and handle the pressure at times,'' he said.
''We don't lack ability. It's just that they are playing tough games every week and we probably have some weeks where we get away with playing soft games and you learn bad habits through that.''
l Leicester, Heineken Cup winners in 2001 and 2002, secured the eighth and final qualifying spot, progressing as a best runner-up. They now face a clash with unbeaten Irish challengers and top seeds Leinster at Lansdowne Road.
Zurich Premiership strugglers Northampton complete the English triumvirate, although an away tie against tournament favourites and twice-European champions Toulouse has installed them as rank outsiders. Biarritz have gained their reward - a home tie against Munster - for finishing top of Pool One above Leicester and fallen defending champions Wasps.
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