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.
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."
He went on: "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."
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.
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."
- Sports & Recreation