A tight first half - played out in high winds and torrential rain in the Welsh capital - saw Gareth Anscombe and Ian Madigan exchange kicks before the Ireland international gave the province a slender half-time lead.
Anscombe levelled things up early in the second period before the defining moment came 20 minutes from the end when wing Isa Nacewa finished off a counter attack against the run of play.
Cardiff Blues hit back with a score from Josh Navidi three minutes from time but despite a brave late rally - that saw a drop goal from Rhys Patchell charged down - the home side could not find another score and had to settle for a losing bonus point.
It was horrendous conditions in the Welsh capital that greeted the teams as high winds and rain lashed the stadium in the run-in to kick-off.
Cardiff Blues started the game going into the fierce wind and early charges up-field from Kristian Dacey and Manoa Vosawai secured a penalty for offside. Despite the wind and rain, Anscombe drilled the kick over for an early lead.
A huge first scrum from Leinster resulted in a penalty back and Madigan nudged his kick over to level the scores after seven minutes.
The visiting set piece looked ominously dangerous, as they won a penalty on home ball and then forced another at the scrum. But on the third attempt the home pack dug deep and won back ball to a huge ovation.
Number Eight Vosawai was offering himself up as an ever-willing carrying option before spilled lineout ball was gobbled up by Ellis Jenkins and taken up to halfway.
It was the forwards who were looking to keep hold of the ball and play a pragmatic tight game into the eye of the storm in the first quarter.
Leinster were dominating territory and an offside decision gave Madigan his second shot at goal but the fly half, released by Ireland this week to play, hit the kick well only to strike the right upright.
Madigan was involved again minutes later as his low cross field kick was collected by Fergus McFadden on the right flank, but after rolling out of a huge last ditch tackle from Patchell he spilled the ball as he went for the line.
The game was beginning to swing from end to end as a superb break from Alex Cuthbert saw him tear up the wing, rattle into contact and then feed Patchell and in turn Lloyd Williams on his shoulder.
The scrum half was felled in full flow by a high tackle from McFadden, that saw the wing yellow carded, but the resulting penalty attempt drifted wide from Anscombe.
The home forwards continued to work tirelessly to defy the conditions with Jenkins winning a penalty, after superb work at the breakdown, as he tried to try and repel pressure.
One last attack in the first half saw a cross field kick from Madigan spilled forward by Nacewa, under pressure from Cuthbert. But the penalty was already secured and Madigan duly slotted it for a 6-3 lead at half-time.
An early penalty at the breakdown in the second half gave Anscombe another attempt at goal and the fly half smashed the kick over, with the wind nearly sending the ball over the stand roof.
Cardiff Blues, though, were dealt a major blow soon afterwards when centre Cory Allen was stretchered off in some discomfort and replaced by Tom James.
Visiting fly half Madigan continued his strong afternoon with his third successful penalty to edge Leinster back ahead.
A new front row was introduced by coach Danny Wilson on 50 minutes, to try and spark from the home pack, and soon afterwards Dan Fish came on as Patchell slotted into the ten jersey.
But, from nowhere, a visiting try shocked the Arms Park as Jack Conan picked up and shot down the short-side before feeding support runner Nacewa to score for a real sucker punch.
The home side responded by introducing Navidi and then marching a driving maul up field to win a penalty that was kicked to the corner. The forwards took ball on but it was then spilled after several phases and the chance gone.
The game was being played at a lively pace and a quick tap penalty from scrum half Tomos Williams caused panic in the Leinster defence and James stepped up and bounced off two tacklers.
A penalty was forced and saw Jordi Murphy yellow carded as Blues kicked to the corner. Finally, on the third attempt, they claimed the ball and drove over, with Navidi the man touch down and Patchell cutting the lead to a solitary point with his pinpoint conversion.
Leinster looked to run down the final few minutes, but when they spilled ball with less than 30 seconds left it sparked one last wave of attack.
The forwards went forwards but Patchell saw his drop goal charged down after he went to the pocket before a knock on signaled the end.
04m Anscombe p 3-0
07m Madigan p 3-3
40m Madigan p 3-6
44m Anscombe p 6-6
50m Madigan p 6-9
59m Nacewa t 6-14
76m Navidi t Patchell c 13-14
Cardiff Blues: 15 Rhys Patchell, 14 Alex Cuthbert, 13 Cory Allen (James 47), 12 Rey Lee-Lo, 11 Blaine Scully, 10 Gareth Anscombe (Fish 53), 9 Lloyd Williams (T Williams 70); 1 Thomas Davies (Thyer 50), 2 Kristian Dacey (Rees 50), 3 Salesi Ma’afu (Filise 50), 4 Jarrod Hoeata (Cook 70), 5 James Down, 6 Josh Turnbull, 7 Ellis Jenkins (Navidi 63), 8 Manoa Vosawai
Replacements: 16 Matthew Rees, 17 Bradley Thyer, 18 Taufa’ao Filise, 19 Macauley Cook, 20 Josh Navidi, 21 Tomos Williams, 22 Tom James, 23 Dan Fish