Cahill was bought by Everton for just £1.5m in 2004 and went on to start more than 200 Premier League games under Moyes, scoring 56 goals.
And the Australian believes the Old Trafford hierarchy have identified one of just two managers capable of grasping what it means to be Manchester United boss.
He told Sky Sports News: "You look at any other manager in the world I think, other than Pep Guardiola, there's only David Moyes that would suit a club like Manchester United.
"He's done so much for the game, and to have the opportunity to go to one of the biggest clubs in the world and continue their level of success is a remarkable measure of the man he is and how much they respect him as a football manager."
Moyes has signed a six-year contract and was chosen despite a relative lack of European experience.
Cahill said: "Do you want that manager that's been at the top level and bounced around all the European clubs, bringing in all his own staff in, changing the footprint of what Manchester United's all about, breaking wage structures even more, signing the big superstars?
"Or do you want somebody that respects the club, respects the production of what they're doing and understands the budget?"