Joe Schmidt's men finished strongly and carved out a last-gasp scoring chance from a scrum on the Scarlets' 22. However, just when the drop goal looked on, Sexton fumbled Eoin Reddan's pass and that was the last act of an entertaining but error-strewn game.
Scarlets fly-half Rhys Priestland opened the scoring in the fourth minute and his hat-trick of penalty goals created a 9-5 interval lead for the hosts.
Eoin O'Malley had profited from a Sexton cross-field kick and touchline pass from Horgan to score his first senior try for Leinster, but a 54th minute yellow card for Rhys Ruddock preceded the Scarlets' one and only try.
Gareth Maule was the player to get the touchdown, crossing in the left corner after some forceful driving from the forwards and some quick handling from the backs.
Priestland, however, missed the conversion and Leinster responded with a 71st minute try from the excellent Fergus McFadden, who scampered past three Scarlets defenders. That five-pointer was followed, five minutes later, by Horgan's dashing run in from the right.
The momentum was with Leinster at this stage, but despite the best efforts of inspirational number 8 Sean O'Brien amongst others, there was no further scoring.
The only Magners League match to survive the cold snap on Friday started with the Scarlets almost registering an early try. Morgan Stoddart forced his way through and was inches away from scoring.
Horgan's long reach saw him touch down just ahead of Stoddart, and confirmation from television match official Hugh Watkins gave Leinster a relieving 22-metre dropout.
Devin Toner, who won his first Ireland cap in recent weeks, was singled out for a lineout infringement and Priestland popped the fourth minute penalty through the uprights for the lead score.
The Scarlets played most of the rugby in the early stages, but Leinster were soon within scoring range. Sexton hacked on a loose pass from Regan King and forced the retreating Priestland into the concession of a five-metre scrum. The Scarlets won possession back after a wheeled scrum and cleared the danger.
Space was at a premium in a crowded midfield and pressure from Gordon D'Arcy forced young full-back Dan Newton to kick out on the full.
Then, in the 14th minute, Sexton's well-placed cross-field kick bounced up for Horgan to gather it and his looped pass back inside sent O'Malley, who was quickly on the scene, over to the right of the posts. Sexton missed the conversion, drawing his kick away from the left hand post.
Tavis Knoyle darted towards the visitors' try-line after Scarlets captain David Lyons had disrupted a Leinster ruck, but again Horgan came to his side's rescue - hauling down the Scarlets scrum half. Leinster survived the ensuing bout of pressure in their 22, courtesy of a strong scrum.
A quick tap penalty and weaving run from Knoyle showed his sense of adventure and two more penalties from Priestland punished Leinster's indiscipline as the Scarlets retook the lead before half-time.
A degree of niggle crept into the game following a high challenge by O'Brien on Newton. Referee Nigel Owens gathered the players together for a stern talking to, before a series of unforced errors brought the half to a scrappy end.
Nigel Davies' charges made a strong start to the second half, with the lively Knoyle willing his forwards through a series of pick and drives and Joe Ajuwa and King both causing problems for Leinster's fringe defence.
The ebb and flow continued with Leinster then enjoying one of their best spells with ball in hand. Centre O'Malley featured twice before Maule was penalised for going offside. However, the straightforward penalty was miscued by Sexton.
Leinster lost Ruddock to the sin-bin for an alleged shoulder-led challenge on Ken Owens as he tried to clear out a ruck. The visitors felt that the match officials' decision was harsh and Priestland rubbed it in by landing the resulting penalty for a 12-5 advantage.
Wales duo Matthew Rees and Stephen Jones were then introduced and the former, having conceded a penalty at a midfield ruck, watched Sexton miss his third shot at the posts.
The Ireland fly-half had his attempted grubber kick blocked by Knoyle and Isa Nacewa had to scramble back under his posts to retrieve the situation for Leinster.
The visitors fell further behind in the 64th minute, the Scarlets profiting from powerful bursts through the middle from replacements Rob McCusker, Deacon Manu and Rees and lovely hands from the likes of Jones and King.
Maule's try looked like being the match-winning score for the Scarlets, until the free-running O'Brien and McFadden inspired Leinster's comeback.
McFadden finished off brilliantly for his try, beating Ajuwa's first-up challenge and two subsequent tackles after a breathless 20-plus phases from Schmidt's side.
The Scarlets' lead was down to seven points and after a Priestland penalty fell short of the target, Leinster manufactured their third try with Horgan slicing through for his 34th touchdown in league history. The build-up to it included a barnstorming run from O'Brien, out of the Leinster half.
Sexton landed the conversion from the right and with props Peter Edwards and Manu both sustaining arm injuries, the Scarlets had to dig deep to stave off Leinster's attempts at snatching a late victory.
|Scarlets Score Card|
|Leinster Rugby Score Card|