David Moyes' were back to their best to all-but end City's hopes of catching Manchester United and maintain their own push for Europe a week after sliding to a 3-0 loss to the Latics.
Gibson said: "I think we needed to do that after the performance last week, but I think we have been great all season.
"Last week was a bad week for us but we have bounced back and showed what kind of a team we really are.
"We all knew as a team that wasn't Everton last week. We had an off-day.
"But we showed everyone what we can do, even going down to 10 men I think we still deserved the three points.
"That is what we are all about, closing down quickly, getting the ball back and then showing our quality when we get the ball.
"We played really well. We had to dig in it at times. They are a great team as well but I think we deserved it."
Everton's cup exit has damaged their chances of qualifying for the Europa League but they still have hopes of earning a place through their league position. The result took them back above neighbours Liverpool and into sixth place.
Gibson said: "It was disappointing going out last week but we have shown what we can do here.
"I think we will kick on from this performance and hopefully get as many points as we can."
Gibson also had praise for team-mate Leon Osman, who is enjoying another fine season and was also made captain with Phil Neville left on the bench.
He said: "He is on fire at the minute. He is showing everyone what a great player he is.
"He had the armband and I think the goal topped his performance off.
"It was nice for 'Jela' (Nikica Jelavic) to get the goal too. It's been a while so I am glad he got one."
Gibson, a substitute against Wigan last week, was pleased to be back in the starting line-up after what has been a frustrating season personally.
The 25-year-old has been limited by a thigh injury for large parts of the campaign and is expected to undergo surgery in the summer.
The Irishman said: "I always enjoy playing here at Goodison and getting three points makes it even better."