A controversial 69th minute penalty from man-of-the-moment Sean Maguire changed the course of the game, and left the Harps bench fuming.
Garry Buckley went down in the box under pressure from Ciaran Coll and referee Paul Tuite pointed to the spot.
Harps were incensed but after all the commotion, Maguire stepped up to slam the ball past Ciaran Gallagher to delight the small band of City fans who had made the long journey.
City had to play the final 15 minutes with just ten men after Steven Beattie got a second yellow card for a rash challenge on Adam Hanlon.
As expected, City pressed hard initially and enjoyed plenty of possession against dogged Harps who made five changes from the side that had started in the defeat against Longford on Friday.
With Stephen Dooley pushing forward with real menace on the left flank, Cork searched for the early score that would have sparked alarm in the home camp.
But Harps defended resolutely and midway through the first half enjoyed a decent spell that led to a couple of corners.
Indeed, Harps carved out the first real chance of note with a Hanlon drive just wide of the target shortly afterwards.
When Karl Sheppard appeared to have outfoxed the Harps defence, Ciaran Gallagher was quickly off his line to intervene and then on 40 minutes Alan Bennett rose to firmly meet a Dooley corner but headed over the target.
City began the second half with renewed vigour and Gallagher was forced into making an acrobatic save to tip over a Gary Buckley shot from the edge of the box.
Sheppard then squandered a chance as City really began to get on top with Harps being increasingly stretched.
But plucky Harps occasionally reminded the Munster visitors that they still had intentions of conjuring up a shock result.
Indeed, on 61 minutes, Ruairi Keating scampered down the right and cut in towards goal before letting fly with a cracking effort that skimmed off the roof of the net.
However, Maguire’s penalty put city into the driving seat.
Sean Houston went fired over the bar for Harps moments later as the home side upped their intensity again.
But City, despite being down to ten men, held out even though Dave Scully had a late chance to salvage an equaliser for the battling home side.
At the start of the campaign manager Ollie Horgan said that to stay up in the Premier Division this season would be an even bigger achievement that getting promotion via the play-offs last November. He could well be right.
Second bottom Wexford Youths host Finn Harps at Ferrycarraig Park on the penultimate weekend of the season. That game could well decide who goes into the relegation play-off.
So Cork march on and with Dundalk juggling between domestic and European commitments, there remain good grounds for believing that the Premier Division title could yet be resting at the end of the season at Turners Cross for the first time since 2005.
Gallagher; Boyle, J. Mailey, Cowan, Coll; Hanlon (McNamee, 75), Houston, Molloy (Harkin, 57 mins), Funston (Curran, 73); Scully, Keating.
McNulty; Beattie, Mulcahy, Bennett. O’Connor; Sheppard (D. Morrissey, 60 mins), Bolger, G. Morrissey (O’Sullivan, 90 mins), Buckley; Dooley (McSweeney, 77 mins), Maguire.
Mr. P. Tuite (Dublin).