The fortunate hosts, who hammered Ulster in the 2012 final, their only previous meeting in the competition, trailed at the break and with the sides level nine minutes from time, had a superb Ross Byrne penalty to thank for the tightest of wins.
Ulster were up for it from the start and the pressure paid off after six minutes when Garry Ringrose was hassled inside his own 22 metre line and a determined combination of Ireland team mates Iain Henderson and Rory Best charged down his clearance to gift Kieran Treadwell the opening try.
The four-times champions were not rattled by the early wave and responded straight away, hammering the Ulster line until flyhalf Byrne, in for the injured Johnny Sexton, found a gap to pull a try back within five minutes.
The visitors suffered another setback moments later as captain Best, set to retire after this year's World Cup, left the field close to tears with an ankle injury.
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But with John Cooney - the best player on the pitch in the first 40 minutes - initially kicking more accurately than Byrne and his Ulster team mates continuing to land some huge hits, the less fancied Irish province inched ahead 13-11 at halftime.
Starting the second half as they did the first, the northern men should have pulled further ahead only for the tournament's joint top try scorer Jacob Stockdale to spill in the corner when he made the costly decision of trying to place the ball down inside the in-goal area instead of diving on top of it.
Leinster were the highest try scorers in the pool stages for a reason - averaging more than four a game - and they ruthlessly took their first opportunity of the half as number eight Jack Conan burst his way through to put Adam Byrne clear.
In a potential World Cup blow for Ireland, Leinster lost Dan Leavy to a serious leg injury and it looked like it might get worse as Luke Marshall, making his first appearance in almost a year, barrelled over for an Ulster side who would not quit.
Cooney missed the relatively easy conversion that would have edged his side ahead, leaving it to a hobbling Byrne to strike the far more difficult penalty effort and set up a semi-final against either Toulouse or Racing 92.
(Reporting by Padriac Halpin, Editing by Ed Osmond)
