Captain Alun Wyn Jones said Wales had their "foot in the door" after getting their Six Nations campaign off to a winning start.
Rob Howley's men saw England and Scotland record victories on Saturday and the Dragons responded with a 33-7 win over Italy in Rome.
But they will need a considerable improvement to threaten next Saturday's opponents England following tries during the final quarter from centre Jonathan Davies and wings Liam Williams and George North as they posted 30 unanswered points in the second period.
Full-back Leigh Halfpenny kicked 18 points as Wales reeled off an 11th successive victory over the Azzurri, who struck first with an Edoardo Gori try that fly-half Carlo Canna converted.
"The first half proved how much of a test it was," Wales' new captain told ITV.
"We started slowly but the character showed. We got our foot in the door after the way results have gone.
"We worked a little harder, kept the ball and we came together in the second half. We will stay on message.
"We are a day behind everyone else so it is a foot in the door."
North added: "I think, the first half, the weather came down and it was going to be a tight game anyway.
"We had to fight to the end, every inch, but we're happy with the performance.
"We know they are a passionate team but we backed ourselves and it showed."
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