Moyes will take over the managerial reigns on 1 July after agreeing a six-year deal with the Reds.
Vidic has been impressed with the job Moyes has done at Everton and the skipper is looking forward to working with the Scot.
"I think he did a great job with Everton," said Vidic. "He is the man who gets his players to work hard and show discipline and I think he did that well with Everton. And in the last few years they've been successful in getting into the top six.
"I think 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.
"I’m looking forward to working with David and I believe I will have the same success I had with Sir Alex."