Ireland should have had a first minute penalty when Stephen Ward's cross from the left clearly hit Konstantinos Stafylidis on the hand but the referee waved on.
And Giovanni Trapattoni's side were made to pay when Georgios Samaras found Holebas, who scored from the edge of the box on 29 minutes.
The Italian's experimental side, which featured Ciaran Clark, James McCarthy, James McClean and Robbie Brady from the start and Wes Hoolahan after the break, competed well for long periods, but were unable to carve out the openings to take something from the game.
In front of a sparse crowd at the Aviva Stadium, where they were battered 6-1 by Germany in a World Cup qualifier last month, Ireland showed plenty of endeavour and at times craft, but they could not find the killer touch in front of goal.
As a result, Holebas' sweet 29th-minute strike was enough to win the game for the visitors and leave Trapattoni's detractors with enough ammunition to maintain their opposition to his continued presence at the helm heading into March's crucial World Cup qualifiers against Sweden and Austria.
But Ireland had strong claims for a first-minute penalty waved away by Israeli referee Eitan Shmuelevitz and defender Stafylidis could consider himself extremely fortunate not to be penalised for his less-than-effective attempt to deal with Ward's deep cross.
Cox went to ground inside the box under Sokratis Papastathopoulos' challenge three minutes later, but the appeals on that occasion were more muted and also ignored, and McClean scuffed a long-range effort wide.
The Republic looked certain to take the lead with 10 minutes gone when full-back Seamus Coleman, whose performance was a real plus for the hosts, crossed for Cox who headed at goal unopposed but he was unable to hit the target with just keeper Orestis Karnezis to beat.
Greece made Ireland pay when skipper Samaras, who had earlier had to leave the pitch for treatment to a head wound, turned Konstantinos Mitroglou's pass into the path of Holebas, who span John O'Shea and thumped the ball into the bottom corner.
Ireland pressed in the second period but could find no way through as Greece coasted to the final whistle and victory with few alarms.
|Republic of Ireland||Team Statistics||Greece|
|0||1st Half Goals||1|
|3||Shots on Target||2|
|7||Shots off Target||4|
|52.4||1st Half Poss.||47.6|
|63.2||2nd Half Poss.||36.8|