The Liverpool goalkeeper, who had made three crucial saves in the first half, couldn't prevent Tasos Bakasetas' deflected shot in the 48th minute. In stoppage time, he made a costly mistake by gifting the ball to substitute Petros Mantalos, who capitalized on the opportunity to secure the victory.
Victory maintained the Greeks' 100 per cent start to the competition and kept them three points ahead of England at the top of League B Group 2.
If the result was ultimately decided by a stroke of good fortune and an error, the win - a fourth over the Irish inside 16 months - was no more than Greece, still revelling in their 2-1 victory over England, deserved on a night when they dominated for long periods.
Ireland, who ran out looking for back-to-back competitive wins for the first time since March 2019 following their success in Finland on Thursday evening, defended stoutly for much of a hard-fought contest, but were eventually undone in front of a partisan crowd of 30,253.
TrendingOn an emotional night, the Georgios Karaiskakis Stadium fell silent ahead of kick-off as both teams paid tribute to Greece defender George Baldock following his death at the age of 31.
Once the game was under way, the first chance fell to Ireland with just four minutes gone when the ball broke to Evan Ferguson off Lazaros Rota, but the striker was unable to hit the target under pressure from the defender as he attempted to make amends.
However, Kelleher, who had earlier seen a Bakasetas shot deflected just wide, had to be at his best to palm away Giorgos Masouras' stinging 11th-minute volley after defender Liam Scales had blocked Vangelis Pavlidis' initial attempt.
Masouras failed to test Kelleher with a tame header from a Christos Tzolis cross but the Liverpool keeper denied Bakasetas with a foot and then claimed Pavlidis' follow-up.
Nathan Collins blocked another Pavlidis shot after Tzolis had cleverly flicked the ball into his path with Ireland coming under intense pressure but remaining stubbornly resilient.
Front two Ferguson and Troy Parrott were seeing little of the ball but the latter did briefly cause panic for Konstantinos Mavropanos when he blocked his attempted clearance to send the ball towards goal, although keeper Odysseas Vlachodimos was on hand to spare the defender's blushes.
By contrast, Kelleher's goal was repeatedly under threat and he had to save with a foot once again, this time to prevent Dimitris Giannoulis from making the most of Tzolis' 29th-minute pass.
Bakasetas tested Kelleher with a skidding 47th-minute effort as the Greeks returned from the break in determined mood, but he was not to be denied and it was he who broke the deadlock seconds later when his shot from the edge of the box clipped Scales and flew into the net.
Ireland found themselves under the cosh with Greece sensing an opportunity to finish the game off and Tzolis scuffed wide from a promising position.
Ireland head coach Heimir Hallgrimsson moved to refresh his embattled side when he sent on Festy Ebosele and debutant Jack Taylor for Chiedozie Ogbene and Ferguson.
Vlachodimos had to tip a 69th-minute Taylor header over amid a flurry of activity around the home penalty area and the Ipswich man saw appeals for a late penalty waved away after his shot appeared to hit the arm of Mavropanos on its way towards goal.
But the contest was over when Kelleher presented the ball to Mantalos, who rounded him before firing into the empty net.
Republic of Ireland midfielder Josh Cullen told RTE:
"After them scoring so early in the second half, from that point on I thought we were the team that looked more likely [to score] and I thought we got more control in the game.
"I don't think the first half was good enough from us, but we tweaked things at half-time and it's just disappointing to concede so quickly after half-time. I think the boys showed great courage and bravery to come back and lift the performance again.
"Ultimately it wasn't quite enough in the end."