Kilkenny held their nerve to deny 14-man Cork a place in the semi-finals, as Dylan Waugh knocking over seven points.
A purple patch midway through the second half was key as Kilkenny recovered from the concession of two quick-fire goals.
A brace from Rory O'Sullivan early in the second half helped the Leesiders into a 3-5 to 1-10 lead.
Kilkenny had led 1-9 to 1-5 at half-time, with a goal from Marc Howard reducing Cork's deficit just before the interval.
O'Sullivan's goals handed the initiative to a side seeking their sixth straight title, but a Michael Grace point revived Kilkenny.
After Cork's Lorcan O'Neill was dismissed, Kieran Bergin bagged a superb solo point to add to his earlier goal before Waugh and Conor Tobin made sure with late frees.
Scorers -- Kilkenny: D Waugh 0-7 (0-6f), C Tobin 0-4 (0-4f) K Bergin 1-1, P Gahan, S Lawless 0-2 each, C Heffernan, M Grace 0-1 each. Cork: R O'Sullivan 2-1 (0-1f); J O'Callaghan 0-5 (0-3f, 0-1 '65), M Howard 1-0, M Vaughan, D Ring 0-1 each.
Kilkenny -- D Walsh, B Murphy, L Kennedy, T Phelan, J Bergin, C Tobin, J Phelan, P Gahan, P Kenneally, C Heffernan, S Lawless, M Grace, D Waugh, K Bergin, G Geoghegan. Sub: R Power for Geoghegan (50 mins).
Cork -- S Shearon; I Coleman, C Smith, P Cronin; G Carroll, L O'Neill, J Harrington; A Gould, C Buckley; J O'Callaghan, R O'Sullivan, M Howard; D Ring, D Lordan, M Vaughan. Subs: J Corkery for Buckley (43 mins), C Higgins for Gould (50 mins), K Tarrant for Coleman (54 mins), M Sugrue for O'Sullivan (58 mins).
Ref -- T Ryan (Tipperary).