Moyes' 10-month reign as manager came to an end on Tuesday after United's hierarchy decided to sack him after a disastrous Premier League campaign, which has left the club languishing in seventh position.
The final straw proved to be Sunday's 2-0 defeat at Everton, a result that crushed United's hopes of qualifying for next season's UEFA Champions League, with fourth-placed Arsenal 13 points clear with four matches remaining.
Upon his arrival at Old Trafford, Moyes disposed of Meulensteen, assistant Mike Phelan and goalkeeping coach Eric Steele.
The Scot replaced the trio with assistant Steve Round, first-team coach Jimmy Lumsden and goalkeeping coach Chris Woods, while former United defender Phil Neville arrived as a coach.
And Moyes' decision to change United's winning backroom staff despite warnings not to, cost the former Everton head coach his job, according to Meulensteen.
"David ignored the solid advice given to him by many of the staff in place at the time and opted to put his own plans in place, which he’s entitled to do but it backfired on him a little bit," Meulensteen told US radio station SiriusXM.
"I always strongly believe that performances and results are a reflection of what is happening behind the doors. That wasn't good enough.
"Don't forget that Moyes was a respected manager over 11 years at Everton, but I tried to warn him: 'You do realise that, with all respect to Everton, you're going from a yacht to a cruise liner, and that’s how big the difference will be?'
"It's everything that surrounds the club – the players, the pressure, the performances. I think he’s underestimated that and unfortunately it's cost him his position."
Meulensteen, who went on to manage Russian outfit Anzhi Makhachkala and Fulham following his departure from Manchester, is stunned by the speed of United's decline.
The fallen Premier League giants are 23 points adrift of leaders Liverpool, while they will be absent from the Champions League for the first time in 19 years.
"It's really too crazy for words that we are talking about this," the 50-year-old said.
"I've been part of the United backroom staff under Ferguson in the last five years that has proven to be the most successful period in the club's history.
"To now have a really poor season, dropping to seventh in the league table, out of the Champions League suddenly after so many years of constant success, that’s probably why the decision has been made.
"Even before Moyes had the job, I still think United had a fantastic squad with fantastic players."
