Their defender Paolo Cannavaro is appealing his six-month ban for failing to report an illegal approach.
Fabio Cannavaro struck the post late on, while Milan Baros was off target with Liverpool's best chance.
The Azzurri will be without captain Fabio Cannavaro for this weekend's showdown with the Juventus defender serving a one-match ban.
Cannavaro left Juventus after the club had been relegated to Serie B following the match-fixing allegations that rocked Italian football.
"My personal history and my career show my respect towards sports and the ethical principles which support it, " Cannavaro added.
Italy captain Fabio Cannavaro broke Paolo Maldini's record for international appearances as he earned his 127th cap in the draw.
Italy's World Cup-winning captain Fabio Cannavaro, Brazilian midfielder Emerson and defenders Lilian Thuram and Gianluca Zambrotta have already departed for Spain.
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"You get stung by a wasp and then find yourself in the newspaper as if you had been doping, " Cannavaro said.
The likes of Gianluigi Buffon, Fabio Cannavaro, Andrea Pirlo and Luca Toni have the week off to rest after gruelling club seasons.
Italy sealed their place at the 2010 World Cup with a 2-2 draw against Ireland on Saturday, despite Cannavaro's absence through suspension.
On the eve of Parma's 3-0 1999 Uefa Cup final victory over Olympique Marseille, Cannavaro was videoed inserting a drip into his arm.
"You get stung by a bee and then find yourself in the newspaper as if you had been doping, " said a furious Cannavaro.
Cannavaro, the younger brother of Italy's 2006 World Cup winning captain Fabio Cannavaro, has started all but one of Napoli's league matches this season.
Marek Hamsik's opener was cancelled out by an own-goal from Paolo Cannavaro, but Cavani popped up with five minutes remaining to seal the win.
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Italy captain Fabio Cannavaro is suspended for the trip to Dublin and his Juventus team-mate Nicola Legrottaglie is likely to play in central alongside another club-mate Giorgio Chiellini.
Cannavaro was winning his 126th cap for his country, equalling the record of AC Milan legend Paolo Maldini, but it was an otherwise forgettable night for the central defender.
Defender Cannavaro won two scudetti with Juventus, in 2005 and 2006, while the side are currently fighting to clinch third place behind champions Inter and their city rivals AC Milan.
"This year we took advantage of the natural expiration of Cannavaro's contract to bring him back to Turin and we are sure that his technical abilities will help Juventus become more competitive".
The Gulf countries have only previously been able to attract soccer stars at the twilight of their career, such as World Cup winning Italian defender Fabio Cannavaro and Real Madrid legend Raul.
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Cannavaro was previously cleared of a similar charge in 1999, when he was videoed inserting a drip into his arm before he went onto to help Parma beat Marseilles to win the Uefa Cup.
Cannavaro, who was suspended for Italy's 2-2 draw in Ireland on Saturday, which guaranteed their place at the World Cup finals in South Africa next year, is preparing to play in Wednesday's home game with Cyprus.
New Zealand coach Ricki Herbert soon replaced Fallon with Wood, who went so close to clinching a winning goal as he turned Cannavaro inside-out before guiding a well-struck low shot just the wrong side of the post.
Italy soon set about exploiting any means they could to try and gain an advantage - Cannavaro accusing Rory Fallon of using his elbow after another sturdy challenge in the air for which the Plymouth striker was booked.
Cannavaro's dismissal in the 26th minute was equally well-timed for the home side, giving them the majority of the game in which to capitalise and they almost did so immediately from the free-kick resulting from the defender's costly foul.
French forward David Trezeguet equalized in the 23rd minute after Bari goalkeeper Jean-Francois Gillet could only block Diego's shot, but the hosts went ahead again just before halftime with a penalty from Paulo Barreto, who was tripped by Italy defender Fabio Cannavaro.
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