The Blues ran out 2-1 winners with Walcott getting both goals.
The former Arsenal man was also delighted at the return of Seamus Coleman.
“The team were talking together and the crowd were behind us,” he said.
“It was a togetherness tonight and that is what was needed going into the weekend [against Arsenal].
“For me personally, to score my first goals at Goodison was brilliant. The home support is fantastic and to get the win for the team is the most important thing.
“It was needed tonight. We had to dig deep in wet, windy and cold conditions. But the guys did really, really well.
“I am an old-school winger who will work up and down for the team – that work rate has always been there and that is what I am about.
“But that link-up play with Seamus was great to see. He was up and down that wing, helping me out, talking and that’s what it is all about.
“I still feel we need to ease him in but if you ask him he will be raring to go for the next game!
“Jonjoe Kenny has been doing really well in his absence, but Seamus is a new animal and I don’t know how he gets that energy!”
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