Having fallen eight points adrift of the Gunners following a faltering start to David Moyes' reign as manager, United have stabilised their position over the last month.
An eight-match unbeaten run in all competitions has boosted morale, even if it has not made much difference to their position in the table.
This weekend's clash offers the chance to land a heavy blow, and, after collecting only one point from those tough early-season encounters with Chelsea, Liverpool and Manchester City, prove the Red Devils will be among the major contenders as usual next spring.
And Pallister is expecting a response.
The former United defender, who was supporting the Manchester United Foundation Chevrolet Trax eBay auction, said: "Sunday is a bigger game for United. The pressure is heaped on them.
"We cannot get away from the fact it has been a middle of the road start to the season.
"But the last few games have been pretty good. We are starting to hit our straps a little bit.
"The players know this is a must-win game. They know how to respond to that.
"They have been through it many times. They know what it takes to win these types of matches."
Not that Pallister expects it to be easy.
Arsenal have maintained a consistency none of their domestic rivals have been able to match over the first quarter of the campaign.
On Wednesday they became the first English team to defeat Borussia Dortmund on their own ground in the Champions League and, in Aaron Ramsey, boast one of Europe's form players.
"Arsenal are playing great football with great style and great panache," Pallister said.
"They are very silky and energetic. It is a brand of football Arsene Wenger has tried to implement for a number of years.
"Purists would go and watch Arsenal because they entertain.
"They are top of the league and rightly so.
"Arsenal have still got their detractors, who don't think they are good enough to win the title. But they are not going to be too far away."
Should the visitors prevail, Moyes' detractors will doubtless be out in force once more.
Not that Pallister feels they should be listened to.
For he is convinced United have appointed the right man to replace Sir Alex Ferguson, and the teething troubles were only to be expected after such a massive upheaval.
"Everybody has been looking for a bit of fallibility in United for a long time," Pallister said.
"The manager has left after 26 years. There is a new man at the helm. Everything is going to be scrutinised and dissected.
"But as I have said many times, we have got the right man for the job.
"Once he gets his feet right under the table and understands more about his team and they understand more about his methods, they will only get better."