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The 20-year-old's second try of the match brought Italy to within a converted try of only a second ever away RBS 6 Nations victory but Leigh Halfpenny eventually kicked Wales out of sight.
It took just four minutes for Cuthbert, last year's Championship top try-scorer, to open his 2014 account - dotting down in the right-hand corner after Rhys Priestland's clever kick through.
It was a sucker punch for Italy, who enjoyed plenty of possession in the opening minutes, if lacking in penetration whereas Wales seized upon their first chance.
Adam Jones was the acting scrum-half and he fed Priestland, whose grubber kick evaded Italy debutant Angelo Esposito on the bounce, enabling Cuthbert to score his ninth RBS 6 Nations try.
Halfpenny nailed the conversion from the right touchline and just three minutes later, Wales were on the attack again.
This time the defending champions came down the left through George North, who chipped ahead and it was left to Italy's covering left winger Leonardo Sarto to hack clear and deny a sea of chasing red.
The Azzurri steadied the ship however and grew into the match, thanks in the main to some impressive long kicking from full-back Luke McLean and scrum-half Edoardo Gori.
Welsh indiscipline also played its part and when flanker Dan Lydiate was pinged in front of the posts for not rolling away on 14 minutes, 20-year-old fly-half Tommaso Allan had the simplest of kicks to open his RBS 6 Nations account.
It was then Italy's turn to attack and but for Sarto's over eagerness, Italy may have been away in the left corner after a clever kick over the top into space vacated by Cuthbert, but the winger went too early and was penalised.
A fine kick down pitch from Priestland gave Wales an attacking lineout, at which Italy captain Sergio Parisse was penalised for pushing Alun Wyn Jones and Halfpenny bisected the posts for 10-3 on 28 minutes.
But back came Italy and a crossfield kick found Parisse challenging with North for the ball on the right touchline. The Italy captain gathered the loose ball and dotted down but his reaction told it all and after checking with the TMO, referee John Lacey ruled a knock-on against the No.8.
Still Italy were heading towards half-time content at just seven points down but Wales struck on 39 minutes when Williams found Jamie Roberts, who finally broke the Italy line before looking back inside to his centre partner who cruised over the line.
Halfpenny converted for a 17-3 lead at the break but Italy hit back with a try of their own just three minutes after the restart. The Azzurri capitalised on a Welsh knock-on, spread the ball wide to Sarto, who kicked ahead for Campagnaro to chase, kick and dive on the ball to score.
Allan missed the conversion however before Halfpenny followed suit with a penalty and on 49 minutes the Italy fly-half was also errant from the tee after Wales came in at the side.
It was a let-off for Wales, who threatened on 55 minutes through the jinking Priestland but Italy continued to carry the fight to their hosts, only to waste an overlap down the right.
Italy were penalised at scrum time on 66 minutes however, and Halfpenny slotted his second penalty of the match to give Wales the cushion they wanted - but the full-back's error allowed the Azzurri back into the match.
Campagnaro read Halfpenny's long pass on 69 minutes, intercepted and streaked under the posts, giving Allan a simple conversion and bringing Italy to within five at 20-15.
But indiscipline cost the visitors dearly as Halfpenny made amends with his third penalty on 73 minutes to wrap up victory for the defending champions.