The goalkeeper allowed a tame shot from Phil Bardsley to slip out of his grasp and over the line deep into extra time with United looking set to seal a Wembley berth, allowing Gus Poyet's men to move ahead on aggregate.
|MAN UNITED LATEST
|7/1||Man Utd are 7/1 with bet365 to beat Cardiff 3-0 on Jan 28|
The result saw United's dismal run in 2014 continue but Fletcher is confident De Gea, one of the club's better performers this season, has the character to respond.
"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 shoot-out 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 tonight [Wednesday]. But he will be OK."