Cork had to dig deep to see off a fired-up Offaly side but in the end Jimmy Barry-Murphy's men had sufficient firepower to advance to the third phase of the hurling qualifiers.
It was a far from flawless display but Rebels manager Barry-Murphy will be pleased that his side had to work hard before easing to a seven-point win.
Cork led by 1-12 to 1-10 at the end of a gripping opening half. Skipper Pat Horgan showed the way with 1-8 in that period, picking off scores from all angles; his marker Conor Hernon lasted just 24 minutes.
The sides exchanged points a couple of times before Colin Egan gave Offaly a huge boost after 10 minutes when he fired to the net after a cross from Shane Dooley.
Cian McCarthy responded with an excellent point as the sides exchanged four points in rapid succession with both revelling in a lot of loose marking.
Cork finally got moving and prised open the Offaly defence after 19 minutes when rapid handpasses from Pa Cronin and Paudie O'Sullivan sent Horgan through to drive the ball into the top corner.
He followed with a superb point, goalkeeper Anthony Nash landed a monster free and Horgan hit a brace to turn a four-point deficit into a 1-10 to 1-7 lead on 28 minutes.
But Brian Carroll stopped the rot, and Offaly made a blistering start to the second half, hitting the front with two quick points from Dooley and one from Joe Bergin before Conor Lehane eased Cork's nerves with a superb equalising point from the right wing.
But Cork struggled to pull away from a dogged Offaly challenge, although it took a fine save from full-back David Kenny to deny Horgan a second goal from a penalty.
However, Offaly struggled to make headway up front against a Cork defence which tightened up substantially in the second half.
The possession eventually started to yield points for the Rebels, and McCarthy and Lehane hit good scores, either side of another massive free from Nash.
That put them 1-16 to 1-13 in front after 52 minutes, and it seemed that Cork would ease their way to victory when McCarthy landed a brace of points from placed balls from distance.
But Derek Molloy, Conor Mahon and Dooley pulled back points for Offaly to leave two between them with nine minutes left, only for Cork to hit the gas for the closing stages and score five points in a row.
Dooley blasted a free to the net to give a hint of a late rally but Cork closed out the deal.
Scorers -- Cork: P Horgan 1-10 (7f), C McCarthy 0-5 (3 '65', 1f), C Lehane 0-3, A Nash (2f), S Moylan 0-2 each; P Cronin, L McLoughlin, S Kearney, C Naughton 0-1 each. Offaly: S Dooley 1-6 (1-3f, 1 '65'), C Egan 1-0, J Bergin 0-3, D Morkan 0-2, S Ryan, B Murphy, B Carroll, C Mahon, D Molloy 0-1 each.
Cork: A Nash; C O'Sullivan S O'Neill, B Murphy; T Kenny, C Joyce, W Egan; D Sweetnam, L McLoughlin; P Cronin, C McCarthy, C Lehane; J Coughlan, P O'Sullivan, P Horgan. Subs: D Kearney for Sweetnam (41), L O'Farrell for Coughlan (43), C Naughton for McLoughlin (54), S Moylan for O'Sullivan (64), E Cadogan for Cronin (69).
Offaly: E Kelly; C Hernon, D Kenny, J Rigney; D Morkan, R Hanniffy, D Horan; K Brady, S Ryan; C Egan, C Mahon, B Murphy; S Dooley, J Bergin, B Carroll. Subs: D Franks for Hernon (24), C Parlon for Carroll (49), D Molloy for Murphy (56), D Mooney for Brady (62), T Carroll for Egan (66).
Referee: Michael Wadding (Waterford).