United stumbled to their second-worst finish in Premier League history in Mourinho’s debut campaign, finishing the season in sixth place.
Only one have United finished lower in the Premier League era, in the ill-fated 2013-14 season under David Moyes , when they ended up in seventh spot following his sacking.
“The win over Ajax stopped a lot of these doubters out there who were saying it was a poor season because United finished sixth,” said Pallister.
“Okay, we didn’t qualify for the Champions League through our Premier League position.
“But we qualified by winning a European trophy and we also won the League Cup, not to mention the Community Shield.
“All in all, quite a few teams will be jealous of Manchester United’s season.
“That was a massive plus from the season.”
Pallister, speaking to United’s official website, added: “Obviously, winning three trophies was fantastic in itself, but the carrot for everybody at the end of the season was getting back into the Champions League.
“It means so much to the club, not just financially but for the fans and players who want to be involved in the biggest competition in the world.”