The home side might have taken the lead after just five minutes when Denis Stracqualursi fired in a header at the near post, but City defender Joleon Lescott produced a goal-line block to prevent a certain goal.
City began to dominate possession as the half progressed, however, and Edin Dzeko almost opened the scoring after 19 minutes with an effort from inside the box, but Johnny Heitinga produced a telling block.
Samir Nasri smashed the cross bar after 36 minutes with a thumping effort as City began to turn the screw, but they could not find a way through a stern Everton defence.
City began the second half in much the same way as they ended the first, with plenty of possession, but they struggled to make any clear cut chances. A Nasri shot after 52 minutes was comfortably dealt with by the Everton defence.
The home side took the lead after 60 minutes and it came courtesy of former Manchester United man Gibson, who thumped a fine effort from outside the area into the top corner.
City pushed for the equaliser late in the half with Micah Richards and Dzeko both going close, but to no avail.
Everton will look to continue their winning run when they face Wigan on Saturday, while City host Fulham.