Blackburn midfielder Andrews' sweet 36th-minute strike and a second from substitute Cox at the death secured a fourth successive win for the Republic and extended Giovanni Trapattoni's unbeaten record against the team he once managed to three games.
Once again, Ireland's industry and commitment saw them through against technically more gifted opponents despite nine changes to the side which defeated Macedonia in Skopje at the weekend.
Cesare Prandelli's men may have dominated possession but they failed to make the most of it and when the chances came their way, the Irishmen took them with some aplomb in front of a crowd of 21,516.
The large contingent of Italian expatriates at the Maurice Dufrasne Stadium in Liege made their feelings abundantly clear on the final whistle on a night when Sean St Ledger led a gritty defensive display in his first game for 11 weeks, while Wolves midfielder Hunt turned in another terrier-like performance.
It may not have had the significance of the Republic's famous 1994 World Cup finals win over Italy at the Giants Stadium in New York, but it served to illustrate a depth to Trapattoni's squad which many of his predecessors have not enjoyed.
He in effect fielded a second XI, having dispatched senior men Shay Given, Kevin Kilbane, Aiden McGeady and Robbie Keane on top of the absent Richard Dunne, Damien Duff and Kevin Doyle in the wake of Saturday night's 2-0 Euro 2012 qualifying victory in Macedonia.
Opposite number Prandelli, who played under the 72-year-old during his first spell as Juventus manager, retained only five of the men who started their 3-0 win over Estonia but still included commanding defender Giorgio Chiellini, midfielder playmaker Andrea Pirlo and rising stars Giuseppe Rossi and Giampaolo Pazzini.
The Italians oozed class on a pitch made slick by persistent rain, with Pirlo and Riccardo Montolivo orchestrating things behind the front two and right-back Mattia Cassani playing almost as an auxiliary winger.
But for all the pressure they mounted before the break, Ireland keeper David Forde, starting his first game for his country, did not have a save of any note to make during the opening 45 minutes.
Pazzini shot wastefully wide from a tight angle with 25 minutes gone after being played in by Claudio Marchisio, and Antonio Nocerino blasted a long-range effort inches over the crossbar two minutes later.
But while the personnel had changed for the Republic, the system and the mentality had not and that will have delighted their Italian coach, who has been trying to instil that message for the last three years.
When they defended, they did so in numbers and with characteristic determination, and if their approach in attack was more rugged than that of their opponents with Shane Long and Andy Keogh causing problems for Chiellini and Alessandro Gamberini, it proved effective.
Long and Keogh exchanged passes on the edge of the box before the Reading man dragged a 13th-minute shot across the face of goal to serve warning, but when the breakthrough eventually came nine minutes before the break, it was from an unexpected source.
Gamberini was penalised for the latest in a series of fouls on Long 25 yards out, and Hunt shaped to curl the free-kick towards goal.
But he instead tapped the ball sideways to Andrews, whose skidding shot was firm and true as it flew past the diving Emiliano Viviano and into the bottom corner.
The goal was greeted with delight by the small band of Ireland fans inside the stadium, but did not go down well with the locals.
Prandelli made a double change at the break as Pirlo and Rossi made way for Angelo Palombo and Alessandro Matri, but the pattern of the first half was largely repeated during the early stages of the second.
Matri shot across goal with 54 minutes gone after Pazzini, who was sent off three minutes into the World Cup qualifier in Bari last April, had expertly turned Cassani's driven pass into his path.
Pazzini and Nocerino departed, as did Kevin Foley and, much to his disappointment, Long as both managers made changes, one in an attempt to rescue the game and the other hoping to close it down.
Italy threw everything they had at Ireland as the clock ran down but substitute Sebastian Giovinco's 86th-minute free-kick, which only just cleared the crossbar, was as close as they came to snatching a draw.
However, their hopes were finally dashed at the death when Cox made ground to get on to the end of Hunt's cross to seal a creditable victory.