The Gaeltacht side rattled the odds-on favourites in the first-half with a direct approach into full-forward Ó Lonáin and opened a three-point gap (1-6 to 1-3).
Shane O’Donovan booted in a third minute goal to hand Courty the perfect start but Ó Lonáin gave their full back line a torrid time.
He was twice denied a green flag, firstly by a brilliant save from Stephen Enright and then a Eoghan Bergin block.
In the 10th minute, Ó Lonáin fed Lorcan Ó Corraoin who belted to the net.
Corner back Shane Ó Cuirrín advanced to finish a flowing move as An Rinn stitched together four unanswered points during the second quarter.
Patrick Hurney slotted six points in the second-half (four frees) as Courty edged ahead. Corey De Róiste secured a share of the spoils for An Rinn with an injury time free (1-10 each).
Ardmore hung on for a 5-5 to 2-12 win over St Saviours in an entertaining curtain-raiser on Saturday. Sean Barron produced a point blank save in the third added minute to deny John Paul Jacob and secure their first win of the campaign.
Ardmore raced 5-2 to 0-6 ahead at half time. Michael O’Neill bagged 2-1 with Tommy Power, David Gartland and Kenny Murphy also claiming majors.
Jacob and Shaun Corcoran led the Saviours’ recovery in the second period while Kieran Murphy and Seamus Byrne had goals.
Kilrossanty defeated Brickey Rangers by 2-11 to 1-7. Tommy Prendergast’s composed finish on 16 minutes separated the teams at the break (1-5 to 0-5).
Despite losing top scorer Paul Whyte to a black card in the third quarter, Pa Cunningham killed the tie five minutes from time.
Emmett Gaffney netted a consolation goal for Brickeys.