Danny Wilson’s side were determined to make amends for last week’s disappointing defeat to Edinburgh but faced a daunting trip to Dublin.
Leinster threw everything at the Blues, who stood up manfully, and the game remained in the balance going into the final quarter.
The hosts led by just seven-points going into the final 13 minutes but they racked up three quick-fire tries through Sean Cronin, Barry Daly and Nick McCarthy as the Blues finally ran out of steam.
Cardiff Blues return to the Arms Park next Saturday, hoping to make it third time lucky in the Guinness PRO14, with Glasgow Warriors to visitors.
Leinster got the game underway and although Damian Welch secured the ball the Blues were soon under the pump after the ball was lost in midfield.
The Dubliners spent much of the next ten minutes piling the pressure on Danny Wilson’s side, who stood up manfully.
Leinster turned the screw at the scrum and twice went close to crossing but on both occasions Lloyd Williams produced a fine covering tackle.
Tom James was unfortunate not to win a penalty at the breakdown as referee Andrew Brace came back for the advantage but Ross Byrne fired wide.
Soon after the Blues cleared and worked their way into Leinster territory after the hosts infringed at the set-piece.
With Shingler pulling the strings James, Rey Lee-Lo and Alex Cuthbert all looked dangerous and came close to breaking through before settling for a penalty to take the lead.
Leinster were dealt a major blow on 23 minutes as Isa Nacewa was sent to the sin-bin but it could, and possibly should, have been red.
The centre caught Cuthbert around the neck with a swinging are but after viewing television replays Andrew Brace ruled that a yellow card was sufficient.
Cardiff Blues cleared and almost instantly struck in attack as Lee-lo sliced through however he was unable to find support with a long pass.
The Blues went through the phases before Brace went back to the TMO for a late hit on Lee-Lo but on this occasion no offence was committed.
Leinster rallied and launched sustained period of pressure as the half hour mark approached. They hammered the Blues with wave after wave of attack but they were met by some ferocious defence.
The home side eventually earned a penalty, which was dinked to the corner and then powered over from the driving lineout.
Byrne converted to nudge his side into the lead and he brought the scores to 10-6 at half-time.
Shingler and the Leinster fly-half exchanged penalties in the opening minutes of the second half and the Dublin-based province soon threatened a try.
Barry Daly and Cuthbert raced after a kick through but the Leinster man fumbled the ball into touch.
The home side continued to exert pressure with Byrne missing another shot at goal before making amends on the hour mark to make it 16-9.
Cardiff Blues continued to fight and they were unfortunate to leave Leinster territory empty handed five minutes later.
Following a neat lineout move which saw Navidi peel down the blindside they went on the attack.
They hammered away and appeared close to a breakthrough but were penalised for taking a player out off the ball.
Cardiff Blues were then dealt a major blow as the ball popped out of a breakdown in midfield and fell straight into the hands of Sean Cronin, who raced home.
Byrne added the extras to put his side out of reach and the Irish province displayed no mercy as they claimed quick-fire tries in the closing stages.