Gibson earned himself the hero status he never enjoyed during his playing days at Old Trafford, where United comfortably beat Stoke 2-0.
Gibson, who joined the Toffees earlier this month, struck on the hour mark at Goodison Park with a deflected shot after being set up by Leighton Baines.
Samir Nasri hit the bar for City but the main talking point of the first half was a bizarre stoppage when a fan ran onto the pitch and handcuffed himself to a post.
United, who now trail City only on goal difference, defeated Stoke thanks to two penalties.
Javier Hernandez opened the scoring in the 38th minute after Jermaine Pennant felled Park Ji-sung and Dimitar Berbatov made it two eight minutes after the break, Jon Walters the culprit this time with a foul on Antonio Valencia.
Swansea thought they had added Chelsea to their list of scalps after former Blue Scott Sinclair gave them the lead in the 39th minute with a brilliant volley, which he refused to celebrate.
Things got worse for the visitors when Ashley Cole was sent off in the 86th minute after picking up a second yellow card but they grabbed a 1-1 draw three minutes into injury time in fortuitous fashion as Jose Bosingwa's cross-shot deflected off Swans defender Neil Taylor and into the net.
Third-placed Tottenham eased past Wigan 3-1 to close to within five points of City and leave Latics rooted to the bottom of the table.
Luka Modric set up Gareth Bale for a 29th-minute opener, and five minutes later the Croatia midfielder got on the scoresheet himself with a precise finish.
Bale then grabbed his second in the 64th minute and James McArthur's deflected shot 10 minutes from time was scant consolation for Wigan.
Andy Carroll marked the anniversary of his £35million move from Newcastle to Liverpool with the first goal as the Reds increased Wolves' struggles with a 3-0 win at Molineux.
The opener did not come until seven minutes after the break when Carroll tucked away Charlie Adam's cross. Craig Bellamy took advantage of some poor defending to double the lead in the 61st minute and Dirk Kuyt made it three 12 minutes from time.
