New signing Sean Lamont got the hosts off to a flyer but the Welsh side trailed 16-10 at the interval after Girvan Dempsey's 18th minute strike.
But Michael Cheika's men failed to kill off the Scarlets who hit back through a second-half touch down from replacement prop Phil John and Rhys Priestland's 70th minute penalty.
As well as missing the presence of six Lions, European kings Leinster were without departed duo Rocky Elsom and Felipe Contepomi. But the Scarlets also had problems of their own during a chaotic close season off the field with at least 12 players packed off through the exit door as well as forwards coach Paul Moriarty and chief executive Stuart Gallacher.
However, there was little doom and gloom for the west Wales region in the sixth minute as Nigel Davies' men flew out of the traps and saw Scottish flyer Lamont power over for the opening try. Scrum-half Martin Roberts provided quick service and former Northampton wing Lamont latched onto Priestland's slick pass to finish with ease.
Priestland added the conversion but Leinster rallied instantly after former Ireland flanker Simon Easterby was caught offside. Jonathan Sexton, Leinster's European hero at Murrayfield in May, made no mistake with the resulting penalty to prove there is life after French-bound Contepomi.
Isa Nacewa had an effort ruled out by the touch judge but Leinster were dominant and the visitors' relentless pressure eventually told at the end of the opening quarter. The forwards, led by skipper Leo Cullen, battered away at the Scarlets try-line before former Bath centre Shaun Berne's long-pass found Dempsey to saunter home for a converted try.
Sexton was then inch perfect with a second penalty from distance in the 22nd minute as the hosts lost their way and direction. Then the Scarlets lost their discipline in a frantic final ten minutes of the opening half. Summer purchases Richie Pugh and Rhys Thomas were sin-binned by referee Neil Paterson for persistent foul play and the use of the head respectively.
Scottish official Paterson was forced to show a third card to Leinster prop Stan Wright before Sexton and Priestland exchanged penalties before the interval.
Leinster kept up their dominance after the break and looked to have increased their 16-10 lead when Shane Jennings fended off the challenge of Lou Reed and Easterby. But the television match official judged the ball had slipped from the grasp of the flanker with the try-line at his mercy. And it proved to be a turning point in the contest as the Scarlets grew in confidence in the final quarter - and took their chance with John's try.
Priestland missed the conversion but the young fly-half made amends with a 70th minute penalty after Leinster's Sean O'Brien became the fourth player to be yellow-carded. Leinster pushed for a winner in the closing stages but the Scarlets clung on for a vital victory.
Scarlets - Tries: Lamont, John; Con: Priestland; Pens: Priestland (2);
Leinster - Try: Dempsey; Con: Sexton; Pens: Sexton (3);
Scarlets: D Evans, S Lamont, R Higgitt (L Williams 54), J Davies, M Jones (capt), R Priestland, M Roberts; I Thomas (P John 61), K Owens, R Thomas (D Manu 61), L Reed, D Day, S Easterby (D Manu 40-42, D Jones 58), D Lyons, R Pugh
Unused replacements: E Phillips, V Cooper, D Newton
Leinster: G Dempsey (S Keogh 70); S Horgan, F McFadden (N Morris 75), S Berne, I Nacewa; J Sexton, C Keane; C Healy, J Fogarty, S Wright (M Ross 42-50), L Cullen (capt), M O'Kelly, K McLaughlin (S O'Brien 42), S Keogh, S Jennings
Unused replacements: J Harris-Wright, D Toner, P O'Donohoe
Referee: Neil Paterson (SRU)