Mourinho takes his side to Old Trafford for Monday Night Football in an intriguing Premier League clash.
Moyes faces a tough task in proving himself a worthy successor to Sir Alex Ferguson but Mourinho has backed the Scotsman to make his mark at the club.
"One has an incredible history there and another one has just arrived," said Mourinho.
"Is that a problem for David? I don't think it is. He's an experienced manager, he has had years and years in the Premier League.
"He works at this moment with better players than he did in the past (at Everton).
"In this moment he has different ambitions than he had in previous years. Now he knows to be successful, he has to win trophies with Man United."
That marks a step up in expectation levels, with Moyes set to be judged more critically than when he was art Goodison Park.
"He did his job very well with Everton, that's why he's at Man United," added Mourinho.
"You have to measure success by the ambitions of the club where you are.
"If (Ian) Holloway keeps Crystal Palace in the Premier League, after bringing Palace to the Premier League again, I think it's a huge success. But he doesn't win the Premier League.
"You have to measure success in relation to the potential of the teams. And David was successful at Everton."
Manchester United v Chelsea is live on Sky Sports 1 HD on Monday at 7pm.