However, the Georgians, who also lost 2-1 to Giovanni Trapattoni’s men in September after being forced to play at a neutral venue, had cause for complaint once again.

Finnish referee Jouni Hyytia astonished virtually everyone inside Croke Park when he pointed to the spot after the ball appeared to hit defender Ucha Lobjanidze’s shoulder.

Keane calmly converted the 73rd-minute penalty and then completed the job when he headed home an Aiden McGeady corner to dash Georgian hopes of a first win in the group.

Cuper said: “My players are angry, especially because no-one could understand the reason why the penalty was given.

“I didn’t see anything and couldn’t see why it was awarded against us. It was strange and had a negative impact on my team.

“I’m not saying we deserved to win, but we didn’t deserve to lose.”

The 36th and 37th goals of Keane’s senior international career were enough to claim all three points on a night when Ireland were not at their best.

They enjoyed their own slice of misfortune when midfielder Keith Andrews had a 25th-minute strike ruled out for offside against Kevin Doyle, although the ball actually went in off AC Milan defender Kakha Kaladze.

But Keane’s contribution eventually got the job done in front of watching Italy boss Marcello Lippi, whose side now lead the group only on goal difference.

Trapattoni said: “Robbie has not played every week, but he played here tonight in front of 50,000 in a World Cup qualifier, and you could see he enjoyed it.

“Robbie was very important. He kept running back to win the ball.”

The Republic entertain Bulgaria in Dublin next month when a third successive home win would send them to Bari four days later brimming with confidence.

Trapattoni will need more from his players in his native country that night if they are to emerge with something to show for their efforts.

However, he insisted he never expected anything other than a battle against Georgia.

He said: “I saw Georgia against Italy and they didn’t deserve to lose. They are a great team.”