Exeter looked to be cruising through as they led 25-6 heading into the final half-hour at Rodney Parade, but Newport battled back and would have reduced the gap to just three points at 30-28 had Tom Prydie managed to convert a Scott Matthews try.
His effort went wide, though, and a penalty try for Exeter with six minutes remaining - their fifth try of the afternoon - ensured the Chiefs of the bonus-point victory and a place in the last four.
Adam Hughes, Fetu'u Vainikolo, Tom James and Sam Skinner were the other tryscorers for the visitors, with fly-half Ceri Sweeney adding two penalties and three conversions.
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Newport responded with four tries of their own - all in the second half - through Cory Hill, Luc Jones, Matthews and Jason Tovey while Prydie slotted over two penalties and two conversions as the Dragons gained two bonus points.
The Dragons were forced into a late change when full-back Geraint Rhys Jones pulled out in the pre-match warm-up and was replaced by Tovey.
However, it was two former Dragons who made the first major impact of the game as Hughes and Sweeney helped Exeter into an early 7-0 lead.
Hughes opened the scoring when he gathered a loose Dragons ball on halfway to dance past one desperate tackle and waltz home from 22 metres unopposed.
Sweeney added the conversion but missed the second he had to take three minutes later when Vainikolo grabbed a corner touchdown after a great Hughes break to the Dragons line.
It was not looking good for the Dragons' faint chances of making the latter stages of the cup but Prydie's first penalty of the day clawed the deficit back to nine points.
Exeter almost scored again through James from a break down the left wing but he was brought to ground five metres short.
The visitors continued to press forward and Sweeney made amends for his missed conversion by landing a booming 50-metre penalty when the Dragons strayed offside.
Sweeney's opposite number Dorian Jones came close to pulling a try back when he booted over the Exeter defence twice, only to be denied by a cover dive in the right corner as the Welshman was going over for the score.
The Dragons did themselves no favours, however, when prop Lloyd Fairbrother was deemed to have infringed at a ruck and found himself in the sin-bin for 10 minutes.
It should have been a signal for Exeter to up their game but credit to the Dragons who restricted the Chiefs to a single penalty when Sweeney booted over from 20 metres to make it 18-3.
Indeed, Exeter were fortunate not to have a sin-binning of their own when Tovey was taken late when attempting a clearance kick.
The Dragons at least had the last laugh in the first half when Prydie booted a penalty two minutes before the break to give his side a glimmer of hope for the second period.
However, that hope looked to have been extinguished when James grabbed another Exeter try five minutes after the re-start to further increase the Chiefs already decent lead.
But it turned out the match was far from over.
After Exeter full-back Byron McGuigan was sin-binned for a late tackle, Hill went over for a converted try to renew Welsh belief.
Skinner's try for Exeter eased the Chiefs clear again, but the Dragons kept things interesting with scores from replacements Jones and Matthews to put the home region within one try of drawing level.
But when the home scrum collapsed, Exeter won the decisive penalty try and that proved too much for Newport to overcome, although Tovey's final touchdown try in the corner did earn them the double bonus point.
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