The Blues were edged out 2-1 by the Red Devils but were the more progressive side in periods at Old Trafford, creating a host of chances against Jose Mourinho’s men.
Andre Gomes, Gylfi Sigurdsson and Theo Walcott all had opportunities for the Toffees in the first half but it was United who broke the deadlock – Paul Pogba slotting home the rebound after his penalty was brilliantly saved by Pickford.
Anthony Martial curled home a second after the break but the Blues refused to buckle.
Bernard and Seamus Coleman went close to reducing the deficit before Marco Silva’s side finally got some reward for their enterprising display on 77 minutes - Gylfi Sigurdsson slamming home a penalty after Richarlison had been brought down by Chris Smalling.
Everton pressed for an equaliser but despite the Toffees' best efforts, United held on.
“We showed character and plenty of spirit and we fought right until the end,” Pickford said.
“We dug in as a group and we almost got something out of the game.
“Even when we were 2-0 down we were still creating chances, so you’ve got to take positives from that. We had a great attitude as a set of lads today – we never say die and we were unfortunate.
“We created some good chances, too. When you come to Old Trafford, you know what sort of atmosphere you’re going to get and I thought we stood up to it very well.”
Pickford produced typically assured, confident performance against United, saving impressively from Marcus Rashford, Martial and Pogba throughout the piece.
The 24-year-old's penalty stop was reminiscent of his thrilling one-handed save from Carlos Bacca in England's last 16 World Cup penalty shootout triumph in the summer.
Pogba embarked on a stuttering run-up before striking his penalty and Pickford - who repelled a spot-kick from Luka Milivojevic in Everton's 2-0 victory over Crystal Palace last weekend - revealed he had done his homework on the Frenchman’s technique.
“I did my research and made the save. Even if it even goes three yards out to his left or right, I’ve got enough time to get up and make the save from the rebound but it wasn't to be."
Next up for the Toffees is a home clash with Brighton & Hove Albion next Saturday.
Pickford is confident the Blues boast the resolve to hit straight back against Chris Hughton’s side.
“As a group, we can’t be too down about the result,” he said. “We were unfortunate not to get a point, we kept going, we kept digging in and we showed a lot of character and determination to keep plugging away.
“We’ll go into the game with Brighton with strong character and that’s when we need to bounce back and get three points.”