The sides were level 17-17 at half time, 23-23 at full time. Michael Moynihan and Dave O'Gorman scored first half tries for Sunday's Well and Padraig McSweeney kicked two conversions and a penalty. Mick Irwin and Eoin Cunningham replied with tries for Dungarvan and Conor Foley added two conversions and a penalty.
McSweeney and Foley each kicked two penalties in the second half to send the game into extra time. Foley won the kicking contest in extra time when he brought his tally to nineteen with two successful penalties, while McSweeney slotted over one to end the game with sixteen points.
The Third Division leaders booked their place in the net round where they must await the winners of the St. Mary's Douglas tie which was postponed due to the weather.
Highfield made the long journey to Nenagh Ormond and came away with a place in the third round after a 21-13 win after extra time. The Woodleigh Park side trailed at half time but tries by Mick Milne and Mick Dillane and a penalty by Darragh Masterson tied the scores at 13-13 after eighty minutes.
Masterson gave Highfield the lead with his second penalty in extra time and Alan Stack made sure of his side's place in the next round with a try.
Fermoy fulfilled their Cup tie with Old Crescent at Rosbrien with a much weakened side and they paid the price as they lost 68-0 to the Limerick side.
They scored twelve tries, six in each half and led 34-0 at the break. Andy Brace scored two tries and a conversion. Sean Cunningham, Sean Monaghan and James Nolan got two tries each and John Jo O'Neill, Eugene McGovern, Jason Tong and Luke Malone one apiece. Paul O'Loughran kicked two conversions and Larry Hanley one.
Old Crescent's reward is a home tie against Mallow in the next round.
Kinsale got a walk over from Killarney and will now face Highfield in Round Three.
The ties between Dunmanway against Bruff and Newport at home to Presentation were also postponed and they along with the St. Mary's Douglas tie have all been refixed for Monday 2nd January.
Also postponed were the Transfield Cup tie involving Richmond and Shannon and the McInerney Cup Final between Bruff and Thomond.
British & Irish Cup: Mosley 0, Munster 29;
Cork Charity Cup Final: Dolphin 28, Highfield 0;
Limerick Charity Cup Final: Shannon 13, Young Munster 25;
Munster Junior Cup: Dunmanway v Bruff; Off
Kinsale v Killarney; Off
Nenagh Ormond 13, Highfield 21(aet);
Newport v Presentation Off
Old Crescent 68, Fermoy 0;
St. Mary's v Douglas; Off
Sunday's Well 26, Dungarvan 29(aet);
Transfield Cup: Richmond v Shannon; Off
McInerney Cup Final: Bruff v Thomond; Off
South Munster Under 19 League Section 1: Clonakilty 5, Ballincollig 5;
Cork Constitution 3, Highfield 19;
South Under 17 League Quarter Final: Highfield 48, Charleville 5;
South Under 17 League Play Off: Bantry Bay 12, Bandon 40;
South Munster Under 15 League: Cobh Pirates 5, Old Christians 24;
Skibbereen 5, Bantry Bay 12;
Under 19 Friendly: Old Christians 10, Kanturk 24;
Under 17 Friendly: Midleton 6, Youghal 5;
Old Christians 17, Kanturk 20;
Under 15 Friendly: Sunday's Well 17, Dolphin 39;