Replacement not used: Martin Moore
Andrew Conway, Andrew Goodman, James Tracy and Fionn Carr scored the tries, while Noel Reid kicked 10 points, as Leinster 'A' were the sharper and more clinical side against Bristol who are fifth in the English Championship.
Leinster 'A' will now go into the draw for the semi-finals where they could be drawn against Munster 'A', Newcastle or Bedford.
The visitors stormed into a 10-0 lead at Bristol, with full-back Conway opening the scoring when he and Goodman were left two-on-one with Bristol winger George Watkins. Goodman moved the ball to Conway who finished from close range after a slick move that epitomised the province's attacking performance.
Goodman went from provider to scorer eight minutes later when Leinster 'A' again quickly moved the ball wide, with Luke McGrath, Cathal Marsh and Carr all combining to set up the simplest of finishes for the centre who scored from a yard out.
Adrian Jarvis got Bristol on the scoreboard with a penalty from 22 metres, before taking the hosts to within four points of Leinster 'A' with a second penalty in front of a 2,408-strong crowd.
And Tristan Roberts, on as a blood replacement, kicked Bristol to within a point of Leinster 'A' from long range as the hosts came back into the contest toward the end of the first half. But Roberts missed from 45 metres with the final kick of the half as Dempsey's men took a 10-9 lead into the break.
Bristol's confidence had been restored by the way in which they ended the first half, and they took the lead for the first time in the game on 48 minutes when lock Mariano Sambucetti capped a period of home perseverance by powering over. Jarvis converted to make it 16-10.
The visitors' response was spirited though, with prop Tracy battling his way to the line to score after good work from number 8 Leo Auva'a in the Bristol 22.
Reid, having missed his first three kicks in a wind-affected game, knocked over the conversion to move Leinster 'A' back in front at 17-16.
A penalty from Reid, awarded for a dangerous tackle by Bristol centre Fautua Otto, extended the advantage to four points, and that became seven when Reid again found the middle of the posts after Bristol were penalised for handling at a ruck.
Jarvis nudged Bristol back into contention with a penalty when Leinster 'A' strayed offside, but again the visitors responded positively and conjured the try of the match.
Reid spotted a gap and broke purposefully into the Bristol 22 before offloading to replacement Tadhg Beirne, who quickly released the supporting Carr and he had the pace to finish clinically. Reid added the conversion to give Leinster 'A' a commanding 30-19 lead.
Bristol bounced back with a try when replacement hooker Ross Johnston was driven over from a lineout with Roberts converting, but Leinster 'A' closed out the victory to reach the competition's last-four for the second successive year.
Referee: Andrew McMenemy (Scotland)
Leinster v Zebre
Leinster Rugby v Zebre Rugby
Zebre Rugby (H)
12/02/2016, KO 19:30
Fri 12th Feb 16
Leinster 29-8 Zebre
W W W L L
W W W L L