The Cardiff Blues flanker scored his debut Test try as Wales ultimately subdued a battling Italian side 24-16 at the Stadio Flaminio.
Warburton said: “It was looking all doom and gloom after the England game, but two away wins is good.
“We go into the Ireland game with a lot of confidence now and it is a home game we must win.”
However, Wales know they will need to play much smarter when Ireland arrive at the Millennium Stadium on March 12.
Captain marvel Sergio Parisse apart, Italy did not possess sufficient quality to capitalise on overwhelming territorial dominance in the second-half, and a ninth wooden spoon in 12 seasons of Six Nations rugby now beckons.
They also missed four kicks at goal with Wales requiring a late James Hook drop-goal to calm their nerves following first-half scores from Warburton and wing Morgan Stoddart, while Stephen Jones kicked 11 points.
Warburton, 22, again underlined his quality as Wales’ long-term openside successor to 98-cap Martyn Williams by delivering another quality display, restricting Italy to touchdowns by Parisse and centre Gonzalo Canale.
Warburton also hailed number eight Parisse’s performance.
Man of the match against Wales two years ago, the Stade Francais star did his utmost to leave the visitors contemplating a third Stadio Flaminio defeat from six visits.
After going eight Tests without a win between June and early February, Wales can now reflect on successive Six Nations away victories for the first time since their 2008 Grand Slam season.
And if they can sharpen their attacking edge while also consistently taking right options, then both Ireland and France can expect to face searching examinations.
Scorers for Wales: Tries: Stoddart, Warburton. Pens: Jones (3). Drp gls: Hook. Cons: Jones.
Scorers for Italy: Tries: Canale, Parisse. Pens: Bergamasco (2).
Italy: L McLean; A Masi, G Canale, A Sgarbi, M Bergamasco; K Burton, F Semenzato; S Perugini, L Ghiraldini, M Castrogiovanni, S Dellape, Q Geldenhuys, A Zanni, R Barbieri, S Parisse.
Replacements: A Cicero for Perugini 40, Perugini for Castrogivanni 71, V Bernabo for Dellape 52, M Vosawai for Barbieri, 72, P Canavosio for Semenzato, 71, L Orquera for Burton, 67), T Benvenuti for Masi, 74).
Wales: L Byrne; M Stoddart, J Hook, J Roberts, S Williams; S Jones, M Phillips; P James, M Rees, C Mitchell, B Davies, A-W Jones, D Lydiate, S Warburton, R Jones.
Replacements: Richard Hibbard for Rees 78.
Referee: Wayne Barnes (England).