However each time, Cork dug in and reclaimed the initiative, especially in the final 20 minutes.
In the second period, that was predominantly down to the graft of midfielders Ian Maguire and Mark Collins, and the driving surges of Conor Dorman from defence.
Allied to that, the sharpness inside of Brian Hurley and especially Colm O'Neill meant that Cork finished the game stronger around the pitch and on the scoreboard in front of over 4,200 supporters.
With 15 minutes left, Dean Rock's eighth point put the visitors 0-15 to 1-10 in front but they would only score once more before the final whistle.
Cork, meanwhile, were finding joy with their inside duo O'Neill and Hurley, and a point each from them levelled matters.
Mark Collins pointed as the game moved into its decisive phase, but Cork's play was now notable for their ability to move the ball from one end of the field to the other with pace. Eoin Cadogan's 65th minute point underlined this, but it was bettered by the last point of the game, fittingly from the man of the match, Conor Dorman, to seal the league points.
Earlier, the only goal of the game in the 12th minute was timely, even if the hosts were only a point down at the time. Dublin were finding their inside forwards well and Kevin McManamon had already shown he had the turn on James Loughrey when required.
However, the tempo of the game shifted when Jamie O'Sullivan's gamble was rewarded with a simple tap-in goal after his defensive colleague Dorman scuffed his effort at goal.
Cormac Costello and Dean Rock responded quickly to level matters at 1-2 to 0-5 after 15 minutes, and the impressive midfielder Shane Carthy and then a Dean Rock free made in four Dublin points in a row for a two point edge.
Again, however, Cork rallied, with their two-pronged inside attack of Brian Hurley and Colm O'Neill, levelling matters with points, and then the raiding Conor Dorman put Cork in front.
It's been quite a week of scoring for the defender, after claiming a fine goal for UCC in their Sigerson Cup win over Cork IT last Tuesday.
The most aesthetically pleasing moment of the first half - which had the purists salivating - was a superb piece of fielding from Denis Bastick in the 33rd minute.
All the better then that he found Shane Carthy who supplied Rock for the point. Level again at 1-5 to 0-8.
O'Rourke, getting into the game more, added a Cork point before O'Neill pointed a free from an acute angle. But their half time lead was halved when Eoghan O'Gara escaped the clutches of Eoin Cadogan to make it 1-7 to 0-9.
Cork had the type of second half start that has coaches tearing their hair out. Misplaced passes and general sloppiness contributed to Dublin edging two points ahead.
It could have been more only that Cork keeper Ken O'Halloran stopped a certain goal by intercepting an Eoghan O'Gara centre.
Brian Hurley almost created a goal chance for Dorman but he failed to get the crucial touch, and at the other end Dublin made Cork pay when McManamon was fouled in the act of shooting - Rock converted the close in free for a 0-13 to 1-8 lead.
However Hurley and O'Neill always gave Cork a potency up front - the Castlehaven man having a great battle with Eoin Culligan.
Scorers for Cork: C. O'Neill (0-5, four frees), C. Dorman (1-2), B Hurley (0-3, one free), J O'Rourke (0-2), J O'Sullivan, E. Cadogan M Collins (0-1 each)
Scorers for Dublin: D Rock (0-8, four frees), E. O'Gara, K McManamon (0-2 each), S Carthy, T Brady, C Costello, M Deegan (0-1 each)
CORK: K. O'Halloran; T Clancy, E Cadogan, J Loughrey; J O'Sullivan; C Dorman, B. O'Driscoll; F Goold, I Maguire; M Collins, J. O'Rourke, C. O'Driscoll; C. O'Neill, D Óg Hodnett, B.Hurley
Subs: P. Kerrigan for O'Sullivan (half-time); J Hayes for Hodnett (46); L Connolly for B O'Driscoll (65); K O'Driscoll for Maguire (67).
DUBLIN: S Currie; E Culligan, R O'Carroll, D. Daly; J Cooper, J. Small, J. McCaffrey; D Bastick, S. Carthy; T. Brady, D Rock, E. O'Conghaile; K McManamon, E. O'Gara, C. Costello.
Subs: D Byrne for O'Conghaile (half-time); B Fenton for Bastick (46); C Reddan for Carthy (54); C McHugh for Costello (56); E Lownes for Cooper (57); M Deegan for O'Gara (68)
Referee: R Hickey (Clare).