The Scot was sacked as manager at Old Trafford earlier this week after a disastrous season.
The champions are currently seventh in the Premier League, and are 23 points behind leaders Liverpool.
Savage told The Mirror: "Roy Keane got it absolutely spot-on - now David Moyes has paid the price, the Manchester United players who let him down should be ashamed of themselves.
"Last season, champions United finished 28 points ahead of seventh-placed Liverpool. Fast forward 12 months and they have swapped positions, with 23 points separating them.
"A 51-point swing in less than a season cannot be down to the limitations of Moyes as manager alone. The players have to accept their share of responsibility because results have been unacceptable and performances, in some cases, shameful."
United host Norwich City this weekend, with Ryan Giggs in charge until the end of the season.