At the age of 32, Eames is the youngest full-time boss in the top four divisions of English football.
"We support his playing style and our team responds positively to his philosophy," chairman Tony Kleanthous told the club's official website.
Meanwhile, defender Bira Dembele, 29, and midfielder Tom Champion, 31, have left the League Two club.
Dembele played 57 times having joined Barnet in 2015, while Champion scored once in 32 appearances last season.
Eames took joint charge of the Bees in December when Martin Allen left to join Eastleigh, winning four of his 11 games alongside Henry Newman.
He then won two of the final four games of the season after Kevin Nugent was sacked as head coach in April.
Eames was previously a coach with the club's development and academy sides before working as Nugent's assistant.
"For us this was the bold choice and the right time to finally allow Rossi to take sole control of our football destiny," said Kleanthous.
Barnet finished 15th in League Two this season, 11 points above the relegation zone.
Barnet have appointed former interim manager and assistant first team coach Rossi Eames as their new head coach.