The Red Devils failed to set up a Wembley final against Manchester City after losing on penalties to the Black Cats at Old Trafford last night.
Just minutes before the end of extra time, Sunderland got themselves back in the game when De Gea fumbled a strike from Phil Bardsley, but Javier Hernandez responded almost instantly at the other end, resulting in a penalty shootout.
"David has been fantastic," Fletcher told MUTV. "In the last 18 months I don't think there has been a better goalkeeper around. He really has been fantastic. He's been a shining light, one of the big positives of this season.
"He's disappointed, as everyone is, but he will be fine and he will show what a top goalkeeper he is. He redeemed himself in the penalty shootout with two saves and we should have done better with our execution of the penalties.
"All goalkeepers make mistakes and it's just unfortunate for David that it's happened to him, but he will be okay."
Meanwhile, widespread reports have claimed that United are expected to complete the signing of Juan Mata from Chelsea today.