They rallied for consolations from Sione Tuipulotu, Ceri Sweeney and Ben Breeze, but Selley rounded off the game with his hat-trick try. It was a disappointing Dragons outing for Percy Montgomery, Rod Snow and coach Chris Anderson, marking the region's first home loss against another Welsh side. The Scarlets will now go on to face the winners of Saturday's Ospreys-Ulster game in next week's semi-final at Stradey Park.
Selley eased through a yawning gap for the first try and Thomas, the Llanelli RFC fly-half and a former Wales Schools and Youth player, converted. The youngster then brushed past Sweeney - the man favoured to wear the Wales number 10 shirt on the summer tour - in midfield.
A desperate Jason Forster tap-tackle brought Thomas down, but a suspiciously forward pass freed Havili for the second try. With half-time approaching, Jason Forster and Phil John ended 40 minutes of niggle as both were sent to the sin bin following a brawl.
The Dragons came out after the break determined to spread the ball, but an attack broke down in the Scarlets half, allowing Havili to pick up and sprint 70m for his second try. Another turnover and a long Dwayne Peel kick saw Evans hack on and control well for the fourth try. Andy Powell was sin-binned for a high tackle, but the Dragons lost ball on the Scarlets' line, allowing a spectacular counter-attack that ended with a 40m Selley run-in. Montgomery was hauled off after an anonymous final display before his return to South Africa, as the home fans streamed from Rodney Parade.
Lions scrum-half Gareth Cooper came on for his first game since February following an ankle injury, along with Canada prop Snow for his last game for the club. That revived the Dragons somewhat and another replacement, Tuipulotu, powered over for their first score. A beautiful Craig Warlow pass sent Sweeney in for a try, and wing Breeze did well to finish another. But the Scarlets had the final word, Thomas chipping over the top, Matthew Watkins gathering and sending Selley under the posts.
"That's been a long time coming, we've let a lot of people down and owed the supporters and the club," said Scarlets captain Simon Easterby. "That's just one good performance, we need two more, and we'd like to be facing the Ospreys next week."
The Dragons' departing Springbok star Montgomery was philosophical. "They just wanted it more than us and played really well," he told BBC Sport Wales. "We made too many mistakes and couldn't get our game going.
"It's sad and unfortunate to end like that, but Newport's been great to me and I've really enjoyed my time here."
Dragons (0) 19
Tries: Tuipulotu, Sweeney, Breeze Cons: Sweeney (2)
Scarlets (23) 49
Tries: Selley (3), Havili (2), G Evans Cons: C Thomas (5) Pens: C Thomas (2), B Davies
Dragons: Morgan, Wyatt, Montgomery, Bryant, Breeze, Sweeney, Baber, Black, Jones, Thomas, Gough, Charteris, Oakley, Owen, Forster Replacements: Tuipulotu for Montgomery (64), Snow for Black (55) Not Used: Richards, Sidoli, Warlow, Cooper, Veater
Scarlets: B. Davies, G. Evans, Watkins, Selley, Havili, C. Thomas, Peel, John, Rees, J. Davies, A. Jones, Wyatt, Easterby, Powell, G. Thomas Replacements: Maddocks for Watkins (48), G. Quinnell for A. Jones (53), Cooper for G. Thomas (68) Not Used: Gravelle, Phillips, Byrne
Referee: Donal Courtney (Ireland)