After a week of soul-searching following their shattering FA Cup quarter-final loss to Wigan, the Toffees roared back to their best to shock City 2-0 at Goodison Park.
It was a game City had needed to win to keep alive their slim Barclays Premier League title hopes but, even after having Steven Pienaar sent off just after the hour, Everton always had the upper hand.
A superb strike from Leon Osman after 32 minutes and a breakaway effort from Nikica Jelavic in injury time secured a win that provided a stark contrast to last week's debacle.
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."
Pienaar was dismissed for two bookable offences, first a late challenge on Gareth Barry and then after catching Javi Garcia with his studs.
City had chances to equalise in the second half, most notably as back-up goalkeeper Jan Mucha denied James Milner and Pablo Zabaleta, but Everton's resolve did not weaken.
Gibson said: "That just shows the team spirit. Most teams, against a quality team like City, would crumble but we dug in, kept our shape well and worked hard for each other.
"Luckily we got the second goal at the end."
Everton's hunger was clear from the start and they were unfortunate to have an early goal for Kevin Mirallas flagged offside.
Osman celebrated his England call-up by firing them ahead with a long-range effort and Marouane Fellaini felt he should have had a penalty when he tangled with Matija Nastasic.
Fellaini was involved in penalty controversy at the other end late on when he was penalised for handling a Carlos Tevez shot. Despite the offence occurring inside the area, referee Lee Probert awarded a free-kick outside.
Aleksandar Kolarov fired into the wall and soon after City broke through Fellaini and substitute Jelavic made the result certain with his first goal since January.
Gibson said: "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 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' 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."