Moyes, who took charge following the end of Sir Alex Ferguson's 27-year stint at Old Trafford, began his reign with a 1-0 defeat to Singha All Star XI on Saturday.
But Beckham, who won nine major honours in his 10 years at United, expects the former Everton boss to continue the club's recent tradition of success.
"It's going to be interesting this year as there have been so many changes throughout the Premier League," he told Sky Sports News.
"Jose [Mourinho] is back and that's exciting. It's going to be interesting, but I still think United will win the league. I'm a Manchester United fan!"
Beckham, who announced his retirement from playing at the end of the 2012-13 season, also believes that advice from Moyes' predecessor Sir Alex will help the Scot acclimatise.
"I'm sure he has had lot of mentoring, he is one of those managers that has always picked up the phone and spoken to Sir Alex," he added.
"It was always going to be difficult job for whoever succeeded Sir Alex because of what the manager has brought to Manchester United, what he has brought to the game.
"But the same players are there and I'm sure a few more will be added over the next few weeks. Being manager of Manchester United is a great job."
Moyes has appointed Ryan Giggs and Phil Neville to his backroom staff at Old Trafford, and Beckham went on to speak of the importance of retaining the club's identity.
"It's great to have a legacy and Manchester United have the biggest," he continued. "Ryan Giggs has been there so many years and all the players look up to him. Phil Neville has gone back, such a talented person and coach.
"Obviously Gary Neville was involved in the club and Nicky Butt is involved with the reserves. So many of the young players respect them and look up to them. It's important to have that."