The defender believes the Scot needed support to bring in signings during the summer but backs his former manager to improve his new club's inconsistent form
United slipped to a 1-0 defeat at home to Roberto Martinez’s side on Wednesday evening, which has left the Premier League champions 12 points behind leaders Arsenal in the table.
Moyes has faced criticism for Marouane Fellaini's £27.5 million arrival from Goodison following the Belgian's subdued displays, but Jagielka insists the Old Trafford hierarchy, who failed in their attempts to sign Leighton Baines, Ander Herrera, Cesc Fabregas and Thiago Alcantara, are to blame.
“I have a lot of time and a lot of respect for David Moyes," said Jagielka.
"I do feel sorry for him. The fixture list wasn't that kind initially. The transfers in the summer, the people who were making those decisions didn't help him out one bit.
"It is nowhere near a big problem but the pressure was always going to be high. From being at Everton, where the pressure was not so big initially, he could then grow into the season.
“You can't grow into the season at Manchester United. You need to come out firing on all cylinders when you look at how Arsenal have started and Chelsea and teams like that. You can't give them too much of a lead.”
Jagielka backs Moyes improve United’s form following the club’s fourth defeat of the season but insists the Scot must be backed to bring in his own signings who can improve the squad.
"He will come under pressure but I am sure he knew that once he took the job,” said the defender.
“He wasn't going to get a quiet life at Manchester United. It was a fantastic opportunity for him.
“It proved what a great job he had done at Everton. I am sure he will roll his sleeves up and get the guys moving in the right direction.
"I am more than sure he will get Manchester United where they need to be. He just needs that little bit more time. But sometimes, time seems a hard thing to get when you are at that top level.
“He is a football man. He has some good people around him.He has some great players in the squad. He just needs to add a few more players that he wants week in, week out.
"Once he gets that sorted and he gets more of his team, rather than a team he has been given, that is the acid test for him, once he gets the players in that he wants."