The 32-year-old has been with the club for seven years, having worked with the academy, youth team and development squads.
He then won two of the final four games of the season after Kevin Nugent was sacked as head coach in April.
Barnet chairman Tony Kleanthous told the official website: "We have thoroughly searched through all the available choices and both James and I concluded that the best candidate was the one on our doorstep.
"We support his playing style and our team responds positively to his philosophy. We want to build on the foundations we have created and think we can provide Rossi with the support he needs to strengthen his management credentials.
"For us, this was the bold choice and the right time to finally allow Rossi to take sole control of our football destiny."
Barnet finished 15th in League Two last season, 11 points above the relegation zone.