Defending champions Ospreys were not going down without a fight and two tries in seven minutes towards the end of the first half had given them hope.
But while Tom Isaacs got a third try for the Welsh side, Cian Healy secured Leinster the bonus point with a fourth try for Leinster to set up a semi-final with Glasgow and end Ospreys' season.
Dan Biggar passed up the chance to kick Ospreys into an early lead when he failed with a penalty attempt on six minutes.
Leinster led 7-0 only two minutes later when Fergus McFadden's kick through took a fortunate bounce and landed in the hands of grateful winger Conway who crossed the whitewash with Jonny Sexton converting despite a strong wind.
With 20 minutes gone Leinster camped themselves on the Ospreys try-line but the visitors stood firm in defence.
However they were powerless to prevent Conway extending the lead with his second try of the game two minutes later after scrum-half Luke McGrath off-loaded out of the tackle to his flying winger and Sexton again added the extras.
Ospreys responded well and a kick to the corner narrowly avoided Joe Bearman before a quick tap-and-go penalty from Biggar saw the fly-half wriggle over.
Biggar converted and the gap closed further three minutes before the break as the ball passed through the hands of the fly-half, Sam Lewis and Bearman before Ben John dived over in the corner.
Sexton responded with a penalty for Leinster on the stroke of half-time to send the home side in 17-12 up at the break.
Leinster picked up where they left off after the interval and Conway collected his own chip ahead to complete a superb hat-trick and Sexton converted with the help of the post.
The Irish side continued to build their lead and a Sexton penalty on 56 minutes made it 27-12 but the Ospreys continued to push on in search of a score which they warranted when Isaacs went over and Biggar converted.
Sexton hit back with a penalty and in a frantic final ten minutes Healy wrapped up the bonus point for Leinster with a try that Ian Madigan converted.
|Leinster Rugby Score Card|
|Ospreys Score Card|