The Spaniard joined the Red Devils in the summer of 2011 and received many critics after a mixed first season in England.
However, De Gea has gone on to be one of the best stoppers in the country this year, and Carroll is certain that the boss is the reason for his change in form.
"He's done well for a young keeper. He's made mistakes, but everybody makes mistakes in football," Carroll told talkSPORT. "The problem is that if you make mistakes at Manchester United, everyone talks about it.
"I think he's kept his head this season. I've watched him and he's made some fantastic saves and nobody has really talked about them. You have to believe in your keeper and I think Ferguson does believe in him. That's what you want as a goalkeeper, that the manager trusts you.
"I think Ferguson speaks to him and I think he's done it a lot this season and it's telling from his performances over the past months."
Carroll won the Premier League and FA Cup during his four years at Old Trafford.