Moyes was unveiled as the fellow Scot's successor on Thursday after Ferguson decided to call time on a 26-year spell at Old Trafford.
Vidic has won five Premier League titles and one Champions League during his time under the retiring manager's tenure, an is expecting more of the same with Moyes.
"I'm looking forward to working with David and I believe I will have the same success I had with Sir Alex," Vidic told the club's official website. "David and Sir Alex share some similarities - they are very passionate, they are both winners and you can see the way they approach the games and how they run things."
Vidic believes Moyes' 11 years of stability at Goodison Park has provided him with enough Premier League experience to flourish at United, despite a lack of silverware during his managerial career thus far.
"David Moyes did a great job with Everton," Vidic continued. "He is the man who gets his players to work hard and show discipline.
"He did that well with Everton, because in the last few years they've been successful in getting into the top six."
Moyes will remain in charge of Everton until the conclusion of his contract at the end of the season before making the switch to Old Trafford and beginning his six-year contract on July 1.