United are enduring a desperate campaign, crashing out of both domestic cup competitions in surprising circumstances and suffering their eighth Premier League defeat of the season at Stoke on Saturday.
That result has left the Red Devils seven points adrift of fourth-placed Liverpool, with their chance of qualifying for next season's Champions League fading fast.
Little wonder some observers have started to question Moyes' suitability to replace Sir Alex Ferguson, even if most match-going fans are sticking with the Scot for now.
Pallister is too, having been convinced Moyes was the right man to inherit one of the biggest jobs in the game, even if he had never won a major trophy.
And, once Moyes gets in the men he wants to strengthen his squad, the former United defender is confident he will prove it.
Speaking at the launch of the 2014 FA Community Awards, presented by McDonald's, Pallister said: "He will get it right.
"I can understand why people are writing Manchester United off but I am not going to change my mind.
"For me David Moyes was the right man to take over the job. I still think that.
"You have to give him time to understand what the club is all about.
"Sir Alex Ferguson said you have to give him time. Everybody understood that.
"A big transition is taking place. It is not something that is going to happen overnight."
Indeed, Pallister has not given up on United's clinching a top four spot, especially since the club record £37.1million arrival of Juan Mata.
"Mata is a great signing," said Pallister.
"He will have a terrific career at Manchester United.
"I am sure once he gets used to the new players and they get used to him, they will create a lot more chance than they have done previously.
"It hasn't gone to plan so far but there are still enough games left to make it a good season.
"We have every chance of getting a top four slot and a good game coming up against Olympiakos in a few weeks with a great chance of getting through to the following rounds of the Champions League.
"Then in the summer I think he will buy more players.
"Manchester United will be back. I am certain of that, vying at the top of the league and looking for trophies."
However, Pallister does acknowledge this season's problems are a new experience for an entire generation of United fans, who have known nothing but success.
He does though. For when the 48-year-old signed for the club from Middlesbrough in 1989, they were in the middle of what would have been their worst season since they were relegated had it not been for that 1990 FA Cup win, after which Ferguson enjoyed so much success.
"I remember my first year at Manchester United," said Pallister.
"At one stage we were a defeat away from going into the relegation zone.
"We got so much stick for being the size of club we were and not winning the championship for 26 years.
"The the senior professionals, Bryan Robson and Mal Donaghy, told the young lads to keep doing their jobs and it would come right.
"Ryan Giggs was one of those young lads then. Now he will be imparting that same experience to people like Adnan Januzaj, Danny Welbeck and Tom Cleverley."