The 71-year-old stepped down as manager in May after winning 13 Premier League titles during 27 years in charge at Old Trafford.
Despite his belief he can sustain Sir Alex's legacy of success, Moyes did admit to feeling nervous at succeeding one of the greatest managers of all time.
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"Anybody would have doubts if they took over this job, but I’ve got to say I feel it’s the right job for me at the right time and hopefully I’ll make it work.
"But there has to be an element of fear that comes with managing a club like Manchester United. It keeps you working, it keeps you focused and helps you try not to take your eye off the ball."
Sir Alex personally chose Moyes to become his successor at United and urged fans to give him their backing, but the former Everton boss wants to step out of his predecessor's shadow and write his own chapter in the club's history.
"It’s my team now," he continued. "I’ve taken over and I’m in charge. It has to be a new era. Whatever we say, my job now is to make my history.
"From day one it’s my team and I have to get on and show that. I’ll use the ex-manager as much as I can because of his knowledge.
"But it’s my team now and I have to take responsibility for that. If you think about Sir Matt Busby’s history and Sir Alex Ferguson’s history – they could do a film about it.
"I have to make sure now that my history and my time is something which the fans and people in the future talk about.
"I don’t want to mention Sir Alex in every conversation I have. But in the same breath I’d hate to think in any way that I was disrespecting someone of his level.
"I will always respect him and Manchester United will always do."
United begin their Premier League title defence with a trip to Swansea City on August 17, before facing Chelsea, Liverpool, Crystal Palace and neighbours Manchester City.