United looked set to progress to a Wembley final on away goals as the final whistle loomed in Wednesday's second leg against Sunderland at Old Trafford.
However, De Gea then allowed a tame shot from Phil Bardsley to slip out of his grasp and over the line, allowing Gus Poyet's men to move ahead on aggregate with one minute of extra-time remaining.
Although Javier Hernandez responded almost immediately to send the tie to penalties, United were ultimately beaten after scoring just one out of five from the spot, with two penalty saves from De Gea proving to be immaterial.
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."
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