The Spaniard was in superb form on Wednesday night, making several crucial saves, to give his side the upper hand against Jose Mourinho's men, and Carrick feels the young goalkeeper handled the pressure well.
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"There is no bigger pressure cooker really than this stage of the Champions League and coming to a place like the Bernabeu," he told reporters.
"He was coming back home to Madrid as well, so there was an extra spotlight on him.
"Some of his saves kept us in the games at times. It just showed what he is all about.”
De Gea has often seen his performances criticised since moving to Old Trafford from Atletico Madrid in 2011, but Carrick has backed his colleague to improve.
"It is never nice when people question you but that is natural at a club like Manchester United. You are always getting questioned and criticised. That is just how it is. That is fine. You have to deal with it," he added.
"He is a top keeper and he is only going to get better. To play at this club at such a young age and have so much on your shoulders, says enough on its own."Another youngster who impressed for the Red Devils in the Spanish capital was striker Danny Welbeck who netted the opening goal, who Carrick was pleased to see on the scoresheet.
"Danny was magnificent. He has all the attributes to be a top player but probably doesn't get the credit he deserves," he added.
"If the team is winning, who is scoring is not such an issue.
"As a Manchester lad, he loves the place. He is genuine and gives everything. He worked so hard again and I am delighted for him."