The game looked out of the Llanelli side's reach at half time, as they trailed 25–6, but two unanswered tries from Martin Roberts and Nathan Brew, a conversion and two further penalties from Stephen Jones pulled the score back to 25–24.
Fortunately for him, after Robert Lewis conceded a penalty in front of the posts, Jones finally gave the Scarlets a two-point win with three minutes left to play.
The Irish side opened the scoring with a penalty in the first minute, but those were to be their only points of the match as the Scarlets tore through the Connacht defence seemingly at will.
A Morgan Stoddart try was sandwiched between Edinburgh's two second half scores, and Martin Roberts crossed with a minute left to play, but it was too little, too late for the Scarlets.
The second half was largely uneventful until the Scarlets made a late resurgence, being awarded a penalty try in injury time at the end of the game.
Following lengthy analysis by the Television Match Official, Simon Easterby was awarded a try just before half-time, to take the score to 17–0 going into half time.
A Stephen Jones penalty 10 minutes into the second half made it 20–0, before former Bristol centre Rob Higgitt scored to effectively put the result beyond doubt.
The match was played at Franklin's Gardens on the Saturday of the following weekend, and started at a blistering pace, with Regan King and Morgan Stoddart scoring tries in the first ten minutes.
They stretched their lead midway through the half, through Sean Lamont and the boot of Stephen Myler, and, although they suffered a yellow card to James Downey, they held off a Scarlet resurgence to win 33–20 and qualify for the semi-finals.