The Londoners took the lead in their quarter-final clash at Old Trafford via Dimitri Payet, but Anthony Martial struck back with the clock running down to set up a replay of the match.
READ MORE | LVG 'disappointed' with Fellaini jeers
And De Gea was impressed with the fight his side showed to keep believeing in the equaliser to claw back a vital goal, declaring his team would battle for victory when they meet again.
"It couldn't be today but after the effort we have the replay," the Spaniard wrote on Twitter. "This team doesn't surrender!"
It couldn't be today but after the effort we have the replay. This team doesn't surrender! pic.twitter.com/2vIPLzlXqg— David De Gea (@D_DeGea) March 13, 2016
His compatriot Ander Herrera is likewise maintaining hopes Louis van Gaal's men can progress to the last four at Wembley, writing: "Everything will be decided in the replay."