The Spaniard saved a penalty from Leighton Baines and produced two superb diving stops in the final 10 minutes to preserve his side's lead as Everton laid siege to their goal.
"De Gea, heroic performance winning @ManUtd 2 points with 3 extraordinary saves. Great to see," former United keeper Peter Schmeichel said on Twitter.
Amid the turmoil of David Moyes's disastrous season in charge at Old Trafford and Louis van Gaal's faltering start as manager, De Gea has clearly established himself as United's number one keeper.
He was heavily criticised after arriving in England in 2011 for 17.8 million pounds from Atletico Madrid, tasked with filling the boots of Edwin van der Sar who was United's best keeper since Schmeichel's heyday in the 1990s.
Always deemed to be a good shot-stopper, De Gea's critics doubted whether he could cope with the physical demands of the Premier League and he showed vulnerability to high crosses.
Opposing managers targeted this weakness and United manager Alex Ferguson dropped him in favour of Anders Lindegard several times during his first season.
De Gea never complained publicly and he played 37 out of 38 Premier League matches in the 2013-14 season, a beacon of consistency as United spluttered to seventh in the table under Moyes a year after being crowned champions.
Van Gaal had no hesitation in handing De Gea the number one shirt and he made his first start for Spain in a friendly against France last month before seizing his chance to shine against Everton.
He became the first goalkeeper to deny Baines from the spot, plunging to his right to prevent the England left back converting his 15th Premier League penalty out of 15.
De Gea was powerless to keep out Steven Naismith's headed equaliser early in the second half but, after Falcao had restored United's lead, the Spaniard acrobatically saved powerful long-range shots from Leon Osman and Bryan Oviedo.
"At the end of the first half he stopped a penalty so that was good," Van Gaal said. "He also did very well in the last 15 minutes.
"I said to him it is always good when a keeper has such games. It shall improve his confidence. He was fantastic."
Everton manager Roberto Martinez was also fulsome in his praise.
"De Gea was outstanding," he said. "There were some incredible saves, two were eye-catching, but throughout he was very strong and secure."
(Editing by Amlan Chakraborty)
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