Having been drawn in the so called Pool of Death, the Swansea-based region were determined to kick-off their campaign with a win.
Dan Biggar opened the scoring but Sean O’Brien crashed over for the match-winning try and Jimmy Gopperth booted the visitors out of reach.
“At key moments we made bad decisions,” said Ospreys assistant Gruff Rees.
“We were not smart under pressure, and they were the opposite. They did the right thing at the right time.
“We worked really hard to try and get some attacking momentum, but there were just some key moments.
“We had a penalty at 13-9 which we took quickly which was unforgiveable, and it meant it went to 16-9 and we were back under the pump.
“We had a couple of opportunities where we could have turned the tables on them, but we did not look after the ball, we were too shallow, too lateral in attack.
“It meant they won the race to the breakdown and they killed us there.”
Dan Biggar opened the scoring after just two minutes at the Liberty Stadium but Leinster quickly demonstrated their attacking intent.
Jimmy Gopperth made the break to set alarm bells ringing but the Ospreys survived unscathed.
Gopperth hit-back for the visitors but the Ospreys piled on the pressure in the ensuing minutes.
Richard Hibbard came agonisingly close and the Ospreys opted for a five-metre scrum after Leinster were penalised.
But the Dubliners turned the screw up front and were able to clear their lines.
And they dealt a major blow on the Ospreys when Sean Cronin put O’Brien over the whitewash after a rampaging Jamie Heaslip break.
Gopperth slotted the conversion and added a late penalty to lead 13-6 at half-time.
The fly-halves continued to exchange shots at goal following the break but the Ospreys were unable to get the breakdown they desperately sought.
Gopperth then compounded their woes by landing a late penalty to deny them a losing bonus-point.
