Holders Derry are out after being being undone by Sligo, winners in 2010 and 2011, at the Showgrounds.
A solitary strike from Raffaele Cretaro was enough for the home side to advance to the semi-finals, with Gavin Peers and man of the match Alan Keane standout performers for the Bit O'Red.
Rovers boss Ian Baraclough felt his side had to work hard to earn their place among the final four.
"It was a tough, tough game. Neither side was giving anything away. Derry made it very tough and, with the difficult weather conditions as well, it was difficult to play the nice, easy on the eye football," he said.
"But we got an opportunistic goal from Raf, and we created a lot of chances."
Derry boss Declan Devine was fuming with the concession of the goal, saying: "I said before the game that the team that defended better would win the match. Unfortunately we switched off for one moment and got severely punished.
"I am bitterly disappointed that we lost, I felt there was very little in it."
Rovers were ahead on 15 minutes, with Cretaro cleverly back-heeling to the net inside the box after Derry failed to deal with Joseph Ndo's free-kick.
It was fully deserved too, with Cretaro linking up effectively with Keane on the right on numerous occasions.
Keane broke into the area inside two minutes but Aaron Greene was unable to get a clean strike from his pull-back when Danny North might have been in a better position for the pass.
Another raid from Keane could have opened the scoring earlier, but the full-back's pass across the face of goal failed to pick out a team-mate.
Derry enjoyed some joy down the right in the early stages, with Patrick Kavanagh and Simon Madden putting a couple of decent balls into the area, though the Candystripes' only real effort on goal in the opening period was a David McDaid shot that ended up high and wide.
Derry enjoyed their best spell after the restart, dominating possession and creating a couple of good chances.
Irish U-21 international Barry McNamee had the best of them, as Duffy's cross evaded Keane in the swirling wind, but the defender was in the perfect position to block McNamee's shot on the line. Rory Patterson had a chance as well, but Peers came across to make a vital challenge.
Rovers weathered the brief Derry storm and had a great opportunity to double their advantage when Keane's cross picked out North unmarked in the area, but the striker was unable to convert and Derry escaped.
And the visitors almost nicked a dramatic equaliser in injury-time, as substitute Ryan Curran crossed into the area, and both Patterson and Duffy were in space at the far post. But they fluffed their lines initially before Gary Rogers came off his line to block Duffy's second attempt and ensure his side went into the hat for tonight's semi-final draw.
Sligo – Rogers, Keane, Peers, Henderson, Davoren, Conneely (Ventre 66), Cawley, Ndo, Cretaro (Millien 74), Greene, North (Elding 81).
Derry – Doherty, Madden, McBride, S McEleney, Osborne (Curran 78), Higgins, Molloy, Kavanagh (Duffy 39), McNamee, McDaid (Rafter 73), Patterson.
Ref – N Doyle.