Despite being the overwhelming favourites going into the match, the Red Rose stuttered to a 18-11 win and failed to register a single try in a match where the Italians dominated the second half.
However, the 22-year-old is under no illusions that his side must perform better against the Dragons.
"We all know in the changing room that a repeat of this won't be good enough," Vunipola told Sky Sports News.
"It kind of feels like we lost the game, I think all the boys are disappointed with how we performed.
"We got the win against a quality Italian side but take nothing away from them as they played out their skins, if we perform like that next week we will have no chance."
England face Wales in Cardiff on Saturday in the title showdown.