United has won only two of its opening six matches in the English Premier League - its worst start to a season in 24 years - and is 12th in the standings.
Much of the spotlight has fallen on Moyes, Alex Ferguson's hand-picked replacement, but Giggs believes the team should take responsibility.
"Sir Alex was a great manager and a great influence on the team and the club. You are going to miss someone like that," Giggs said. "But I don't think that is an excuse when individuals aren't playing as well as they can. We are not playing well as a team.
"We haven't got injury problems as we have had in the past. There is no excuse."
United was held to 1-1 by Shakhtar Donetsk on Wednesday in the Champions League and Giggs came off the substitutes bench to make his 145th appearance in the competition, a tournament record.
Accolades for the 39-year-old however, were in contrast to United's form, which has seen the team win only one of its last four games overall.
A shock 2-1 loss to West Bromwich Albion at home in the Premier League last weekend came six days after a galling 4-1 defeat to bitter rival Manchester City, meaning United trails league leader Arsenal by eight points.
"It hasn't been a great start to the season but the good thing about that is we have plenty of time to turn it round," Giggs said.
"We know what this club is like, it is ups and downs and we don't get carried away if we are flying and we don't get carried away when we are not playing well."
Ferguson famously endured a difficult start to his tenure at Old Trafford and after finishing 11th in his third season in charge, many supporters called for him to be sacked. United's hierarchy stood firm and Ferguson went on to lift 38 trophies in 27 years.
Moyes is counting on the club to show similar patience for him and is adamant he is the right man to lead United forward.
"Manchester United is a good club, I am sure they know they picked the right man for the job. Sir Alex was part of that process as well," Moyes said on Friday.
"It's a long season. We'll hang in there and get ourselves as close to the top as soon as we can."
On Saturday, United play bottom club Sunderland, which recently sacked manager Paolo Di Canio but showed signs of revival last weekend against Liverpool. Kevin Ball has taken temporary stewardship of Sunderland.
"Whether he gets the job or not nobody knows," Moyes said, "but Sunderland look as if they played quite well in the last couple of games."