United went down 2-1 at home to Swansea City in their Premier League opener at the weekend and have been criticised for their inactivity in the transfer market.
However, Pallister believes that Louis van Gaal's side will respond against Sunderland on Sunday.
"One result doesn't make a season and there is certainly no need to panic. We saw enough in pre-season to expect a vast improvement on last season," Pallister said at an event for Chevrolet's 'What do you #PlayFor?' campaign.
"On paper we have an easier start to the season than last year and I expect United to bounce back at Sunderland on Sunday. There is time to get it right before we start meeting the big guns, but, equally, you don't want too many results like Saturday and be falling behind before we reach that stage."
Marcos Rojo is reportedly on the verge of joining United from Sporting Lisbon.