A 16-point haul from captain Chris Paterson helped Edinburgh to their first ever win at Stradey Park.
Scarlets (10) 15
Tries: Davies (2)
Edinburgh (8) 21
Tries: Paterson, Blair
Pens: Paterson (3)
Barry Davies got the Scarlets off to a good start with a try, but their new look side could not secure possession.
Paterson crossed before the break after Marcus Di Rollo created a gap, and Mike Blair's 57th-minute breakaway try helped Edinburgh to an 11-point lead.
Davies got his second score to give the home side hope and a bonus point, but desperate final attacks were repulsed.
Both sides took time to settle into the new Celtic League season, before Paterson opened the scoring.
Bruised Scarlets tight-head John Davies - pressed into action following a car crash earlier in the week - was penalised for dropping a scrum, and Paterson landed the easy kick.
The home side's backs were looking dangerous and new fly-half Mike Hercus was given good ball from a short-range scrum.
He combined well with Regan King who fed Matthew Watkins, the Wales centre sending full-back Davies over for the first try.
Another dangerous attack resulted in Paterson's sin-binning as he went off-side to deliberately kill the ball.
But 14-man Edinburgh held out and, with their skipper back, struck moments before half-time.
Di Rollo broke effectively in midfield, allowing Paterson to pick up quick ball off the floor and crash over.
The Gunners then regained the lead with a Paterson penalty early in the second half as the Scarlets squandered possession, especially at the line-out.
The Scarlets' backs were dangerous when given possession
It got worse for the home side as Dafydd Jones lost the ball in contact, Blair picking up as it squirted out of the side of a ruck to run in unopposed from half-way for a converted try.
Another Paterson penalty made it 21-10, but with the clock ticking down Davies' second try gave the Scarlets hope.
After a series of forward drives, scrum-half Clive Stuart-Smith broke down the blind-side and fed Davies, who had the pace to get round Simon Webster.
The Scarlets threw everything into the closing minutes, but the Scots' defence was stubborn.
"We'd planned for this game all season and hopefully it sets us up for the rest of the league campaign," said Edinburgh's man-of-the-match Blair.
"There's a lot of confidence in the side, we're enjoying playing a good brand of rugby together.
"The Borders and Glasgow will also be looking for better efforts, and hopefully the Scottish sides will be challenging in the league this year."
Disappointed Scarlets captain Vernon Cooper vowed to put his side's problems right.
"We've trained well, but that game was a big let down and having a lot of new boys is no excuse," he told BBC Sport Wales.
"To turn over so much possession was crippling, you can't do that against a team like Edinburgh.
"Last season we had a terrible start and we don't want that this year. We have to pull together and put this right."
Llanelli Scarlets: B Davies, James, Watkins, King, Selley, Hercus, Stuart-Smith, I Thomas, Hawkins, J Davies, Cooper, Louw, D Jones, G Thomas, Popham.
Replacements: Byrne for Watkins (71), Bowen for Hercus (59), A Jones for Louw (66), Quinnell for D Jones (66). Not Used: Williams, Yelland, L Davies.
Edinburgh: Leonelli Morey, Paterson, Di Rollo, Jorgensen, Webster, Godman, Blair, Jacobsen, Hall, Smith, Kellock, Murray, Mustchin, Cross, Hogg.
Replacements: Southwell for Jorgensen (40), Hodge for Godman (61), Dickinson for Smith (61), Pringle for Murray (78), Strokosch for Cross (68). Not Used: Kelly, Lawson.
Referee: Alan Lewis (IRFU).