A contentious Kevin McHugh penalty was the main talking point as the Donegal side advanced into the last eight of the competition for the first time in 10 years.
Ramblers dominated the first half an hour but the game swung on a poor refereeing decision as Andrew Mullally awaded Harps a penalty after a Craig Burns tackle on Kevin McHugh.
McHugh himself seemed surprised to be awarded the penalty as Burns won the ball but the veteran striker showed no mercy as he tucked away the penalty.
Minutes later the hosts were two up when Lee Toland fired a free-kick into the bottom corner after a Brendán Frahill handball on the edge of the box.
Further goals in the last 10 minutes from Paul McVeigh and Michael Funston completed a miserable night for Dave Hill’s side.
The home side made three changes from the 1-1 draw last weekend with Hill bringing Brian Fitzgerald, Craig Burns and James Carson back into his side.
Peter Hutton made two changes with Josh Mailey and Seán McCarron starting for Finn Harps.
Cobh started really well and could have taken the lead in the opening quarter when Alan O’Flynn latched onto a ball over the top but the young winger was unable to find the target.
The same player was off target again moments later after he was set up by Roy Kenny, then Davin O’Neill was bundled over in the box but referee Mullally waved away the appeals as Ramblers began to dominate.
Finn Harps’ first real sight of goal came in the 22nd minute when Michael Funston broke free down the right but his cross was missed completely by McCarron who should have done better.
O’Flynn had a header saved by Ciaran Gallagher as Ramblers searched for the opening goal but it was the visitors who took the lead against the run of play with McHugh’s penalty on 34 minutes and Toland’s free-kick four minutes later.
The visitors comfortably saw out the half with Ramblers stunned by the concession of two goals in such a short space of time.
The second half started as the first ended with few incidents of note and the game was over as a contest in the 66th minute when Cobh went down to 10 men after Frahill saw red for a second bookable offence.
Shortly after they were reduced to nine when Kenny was shown a straight red for a poor tackle on James Doherty as the hosts completely lost their discipline.
McVeigh made it three with ten minutes left when Gallagher punted a long ball downfield and the substitute easily slotted past Hallahan.
Funston scored a fourth in injury time when he was played in by Josh Mailey to complete Ramblers’ misery and seal his side’s progression into the quarter finals where they will host Drogheda United.
The good news for Ramblers is they get an immediate chance to exact revenge as they host Finn Harps again on Saturday night in the league.
Shane Hallahan; Brendan Frahill, Craig Burns, Brian Fitzgerald, John Meade(C); Alan O’Flynn (Andrew Bhandarkar 59), James Carson, Roy Kenny, Martin Deady; Davin O’Neill, Kynan Rocks.
Subs not used: Robert Birdsall, Eoin Hastings, Daniel Santry, Jonathan Spillane, Sean O’Mahony.
Bookings: John Meade(34) Craig Burns(34) Brendan Frahill(37+66) Kynan Rocks(44) James Carson(90)
Sent off: Brendán Frahill(66) Roy Kenny(73)
Ciaran Gallagher; James Doherty , Josh Mailey, Keith Cowan, Ciaran Coll; Michael Funston, Shaun McGowan, Mark Forker(Pat McCann 86), Lee Toland; Sean McCarron(Paul McVeigh 62), Kevin McHugh(Michael Doherty 87).
Subs not used: Jordan Duffy, Gary Merritt, Aaron O’Hagan
Bookings: Keith Cowan(23) Sean McCarron(56)
Sent off: None
Referee: Andrew Mullally (Waterford)
Report by Sean Geary