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.
The 73-year-old, finally bowing to mounting pressure, handed youth a chance as his side brought an end to their 2012 campaign and tried to do so in style against the national currently ranked 12th by FIFA.
Ireland slipped significantly in FIFA's ranking table - to 36th place, 24 behind the Greeks - in the wake of their humiliation at the hands of the Germans, and the Italian's response was to include Aston Villa defender Clark, Wigan schemer McCarthy and wingers Brady and McClean in his starting line-up.
He was also forced to make a late change when goalkeeper Keiren Westwood withdrew with a groin injury and Millwall's David Forde was handed an unexpected fourth senior cap.
It all started relatively promisingly for the home side with McCarthy finding space in the middle of the field to feed Brady and McClean on the flanks, and the movement of strikers Shane Long and Simon Cox troubling the Greek defence.
Ireland had strong claims for a first-minute penalty waved away by Israeli referee Eitan Shmuelevitz and defender Konstantinos Stafylidis could consider himself extremely fortunate not to be penalised for his less-than-effective attempt to deal with Stephen 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.
But the Republic looked certain to take the lead with 10 minutes gone when full-back Seamus Coleman, whose combination with Brady proved a repeated threat down the right during the opening 45 minutes, collected the winger's return pass and crossed.
Cox had found a yard of space and climbed to meet the ball unopposed, but he was unable to hit the target with just keeper Orestis Karnezis to beat.
Greece had offered little in response, but gradually worked their way into the game and called Forde into action for the first time when Sotiros Ninis took aim from 25 yards and forced a diving 23rd-minute save.
However, the Republic failed to heed the warning and fell behind six minutes later when skipper Georgios 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.
Long and Clark were both booked for clumsy challenges as the game unfolded in anything but friendly fashion, but the home side had rather lost their way.
They might have levelled in injury time when Brady tested Karnezis with a left-foot strike after being set up by Long, but the goalkeeper was equal to the task.
There was warm applause from the sparse crowd when, on their return, Ireland were joined by in-form Norwich star Hoolahan, winning just his second cap, a fact which had not gone unnoticed by Trapattoni's critics.
Brady departed along with Long to accommodate his arrival and that of Kevin Doyle, who was asked to play a lone striking role ahead of Hoolahan with Cox dropping in on the left and McClean moving to the right.
Ironically Glenn Whelan, the man who had pleaded with his manager earlier in the week to field an extra midfielder, was back in the dressing room having limped off injured before the break.
However, it was McCarthy who almost dragged the Republic back into the game two minutes into the second half when Karnezis failed to hold his long-range shot, although Papastathopoulos prevented Doyle from converting the rebound and Cox's follow-up was blocked.
But while perspiration was not in short supply, inspiration was and the Irish failed to make the most of the possession they were allowed as the game passed the hour-mark with Greece's narrow lead intact.
Ward hacked a 69th-minute shot harmlessly across the face of goal after the Greeks had failed to clear a McClean free-kick, and Greece coasted to the final whistle and victory with few alarms.