The Blues won 1-0 thanks to Kiernan Dewsbury-Hall's strike, with the visitors also playing for the majority game with ten men after Idrissa Gueye's dismissal for clashing with teammate Michael Keane.
Pickford made a hatful of wonder saves to maintain the clean sheet and said afterwards: "It was a big win.
"It's tough to come here and get a result – I think it's the first win for me here as a player – so it feels nice. To dig in and get a result like that felt really good.
"I didn't see (the red card incident) initially but Idrissa has apologised afterwards and it's forgotten about. We focus on the win and we're not going to dwell on little things like that – we focus on the win and enjoy that.
"There are moments in games where you've got to dig in, roll your sleeves up. That's what we did. There are moments in training during the week that you play scenarios where teams are a man down.
"It was a great win for us, the subs came on and made a massive impact – credit to everyone involved."
On his own performance, Pickford also said: "I felt in the moment all game and I work hard every day on the training pitch to put in performances and saves like that.
"To put in that performance and get a clean sheet – another one, which makes it back-to-back now – is the Everton way.
"The lads kept me out of trouble sometimes and I get them out of trouble other times – that's part of a team sport."