Danny Wilson’s side led 13-3 at the interval at BT Sport Cardiff Arms Park thanks to Gareth Anscombe, who claimed a converted try and a penalty.
Leinster fought back after Rhys Ruddock powered over and Johnny Sexton kept the scoreboard ticking in front of a bumper 8,289 crowd.
The Blues were continually frustrated and fell on the wrong side of the penalty count as they were left with a losing bonus-point
Following a cagey start it was the Blues that produced the first try-scoring opportunity. Kristian Dacey received the ball after a clever kick through and made good yards before being hauled down by the cover defence.
Possession was maintained but time was of the essence, as Leinster’s defence was on the back-foot, but the home side were unable to move the ball wide quick enough.
While Cardiff Blues had the first chance, Leinster looked dangerous every time they had the ball and were making good inroads with hard running lines.
It looked as though they had claimed the first try after they worked an overlap on the left which put Rhys Ruddock into space.
Matthew Morgan came across and put his body on the line, but it was the intervention of Blaine Scully that saved the day, with the American international hauling Ruddock’s non-carrying arm into touch before he could score.
However Leinster remained camped in Cardiff Blues territory for the next five minutes and following a sustained period of pressure, Johnny Sexton knocked over a simple penalty for a 3-0 lead.
Sexton was given the opportunity to stretch the Dubliners’ lead further on 24 minutes after receiving a scrum penalty but he pushed his effort wide.
Having already lost Sam Warburton to an early head injury, the Blues were dealt another forward blow after George Earle was unable to carry on after battling a leg injury.
However the Blues were finally awarded a penalty of their own and Anscombe made no mistake from distance to level the scores.
Leinster continued to dominate territory and possession but Anscombe was beginning to work his side into the Leinster half with some fine kicking from hand.
The Blues then took the lead shortly after the half-hour mark when Macauley Cook fielded a Leinster clearance and the ball was worked to the Wales fly-half.
He sliced the Irish defence in two on the counter attack and displayed just enough pace on an arcing run to cross the whitewash for a stunning solo effort.
Anscombe converted his try and soon added a long-range penalty to open up a 10-point advantage.
Suddenly it was the Blues in control and they came agonisingly close to creating a second try after Matthew Morgan and Tom James combined on the counter-attack.
James found Lloyd Williams in support but the scrum-half was unable to finish as Leinster scrambled.
And despite going through numerous phases and hammering the try-line they failed to cross.
But while the Blues were unable to turn pressure into points they passed a final defensive shift with flying colours.
A clever Sexton cross-kick was collected by Dave Kearney and alarm bells were ringing but Scully brought him down and the Blues weathered the storm to lead 13-3 at half-time.
Wales’ Capital Region were dealt a further double-blow with both Taufa’ao Filise and Matthew Morgan unable to come out for the second-half.
Sexton quickly got the scoreboard ticking and a razor sharp break from Noel Reid put the Blues on the back foot.
Wilson’s side escaped unscathed but after receiving several penalties from referee Dudley Phillips, Leinster powered over following a driving lineout and consecutive pick and drives with Ruddock the scorer.
Sexton converted the try to tie the scores a 13-13. As the half progressed the home side began to exert some forward dominance at scrum-time.
That forward muscle laid the platforms for a shot at goal but Anscombe was unable to bisect the posts with the difficult penalty.
They continued to threaten with Williams and namesake number eight Nick Williams combining to good effect however errors or the whistle of the referee continually crept in.
Sexton booted his side into the lead on 70 minutes after Josh Navidi was penalised and sin-binned for a lazy retreat.
The Blues had the opportunity to take a shot at goal to level the scores on 73 minutes but opted for the corer.
They won the ball but Leinster defended the driving lineout well and forced a knock-on as Cardiff Blues went round the corner.
Wales’ Capital Region fought tooth and nail to the death and had several opportunities in the closing stages but yet again they were penalised as the chance was lost.