Moyes was sacked on Tuesday after a disastrous 10-month spell in charge, with United now certain to finish outside the Premier League's top four for the first time.
Midfielder and assistant coach Ryan Giggs placed in temporary control for the final four league matches of the season while United search for a permanent successor.
Meulensteen, among a group of trusted lieutenants of former boss Alex Ferguson who Moyes removed in favour of bringing in his own backroom staff, said Moyes had paid the price for ignoring the blueprint of success forged under Ferguson.
"United were a very successful team, with many successful years behind them. The strategies in place worked. But David ignored the advice that was given to him by many of the staff in place at the time," the Dutchman told CNN.
"He opted to put his own plans in place, which he was perfectly entitled to do, but I think it backfired on him.
"I always strongly believe the performances and the results are a reflection of what is actually happening behind the door and that wasn't good enough, as simple as that."
Meulensteen said he had tried to impress upon Moyes that Everton, the club he was leaving behind, was not on the same scale as the 20-times English league winners and three times European champions.
"Don't forget, David was a respected manager in the Premier League who worked for 11 years at Everton in a very good way.
"But I did warn him: 'Do you realise, after everything at Everton, you're going from a yacht to a cruise liner? That's how big the difference will be'.
"It's not just necessarily the work on the pitch. It's everything that surrounds Manchester United, the players, the performances, the pressure, the style, the identity.
"And I think he underestimated that. It's always easy in hindsight but unfortunately it's cost him his position."
Meulensteen went on to manage Fulham in a three-month spell this season before he was replaced by current boss Felix Magath in February after a lengthy run without victory.
That followed an even shorter spell, 16 days, in charge of Russian side Anzhi Makhachkala last year.
Meulensteen, though, said it was wrong that he, Mike Phelan and Eric Steele were let go by United.
"It's too crazy for words. I've been part of Manchester United's backroom staff for the last five years for Sir Alex Ferguson and the last five years have been proven to be the most successful in the history of Manchester United," he said.
"Now they've had a poor season, they have dropped to seventh, out of the Champions League and suddenly, after so many years of constant success, we're now asking whether Manchester United are good enough to get in the top four.
"To me, there should be nothing else and that's probably why the decision has been made because they want to be back (in the Champions League) and next year they want to be challenging for the league title."
Meulensteen's compatriot Louis van Gaal is the favourite to succeed Moyes, with Real Madrid boss Carlo Ancelotti and former United defender and now Paris St Germain coach Laurent Blanc also linked to the role.
United host relegation threatened Norwich City on Saturday.
- Sports & Recreation
- Premier League
- Alex Ferguson
- Manchester United