United lost 3-1 to Palace at Old Trafford to record their heaviest opening day defeat for 25 years and afterwards Shaw pleaded with the club to give the squad more help before the transfer deadline on Oct 5.
"We have a very good group, but personally I think we need more players to strengthen the squad," Shaw said.
"It can give us a boost. When you look around at how other teams are strengthening their teams, then we must also do it to keep up with the others."
United signed £40 million Dutch midfielder Donny van de Beek from Ajax but have been left trailing by Liverpool, Chelsea and Tottenham, who have all made significant signings. Sources have told ESPN that the club are in talks over a deal for Porto left-back Alex Telles and they have still not given up hope of landing top target Jadon Sancho from Borussia Dortmund.
Away from the transfer market, manager Ole Gunnar Solskjaer has also encountered problems preparing his players for the new season after a short turnaround and Shaw believes a lack of time on the training ground proved decisive against Palace.
"Personally from my point of view the preseason that we've had is not the correct way it should have been, we've had a week all together," Shaw added.
"We haven't been able to prepare in the way that we wanted and that's not an excuse because we have the quality of players to be able to win games like this but we weren't up to it and we've been punished.
"It's the start of a new season, we should have been fresh."