Smith admitted his players would celebrate their derby victory over Harlequins last weekend with a few beers but insists they are set up nicely for Friday's visit of Prato.
"The boys have had a self-imposed drinking ban for the past six weeks," said Smith.
"That is how much it means to the boys to do well. They made a commitment to each other. It was nothing to do with the coaches - it was led by the senior players.
"We will get a lot of belief from the win over Harlequins and it will fuel the fire for our next game."
Prato returned to winning ways themselves at the weekend with 21-3 victory over Golden Flame Rome in Italy's National Championship of Excellence.
Australian full-back Toby Browne kicked four penalties in the win and the former Randwick star has been in superb form with the boot this season.