Sterling went down under a challenge from John Stones in injury time but referee Roger East gave nothing and City dropped two points in the Barclays Premier League title race.
City had dominated proceedings on a wet and cold night but were continually frustrated by the dogged visitors, for whom goalkeeper Tim Howard was outstanding.
City striker Sergio Aguero also had a couple of penalty appeals turned down as City, a week after losing at Goodison Park in the Capital One Cup, once again found Everton a tough nut to crack.
The major talking point prior to the game was the traffic problems on the motorway network around Manchester which delayed many supporters
The ground was not full at the scheduled 7.45pm start and the Everton team had even arrived later than planned, although not severely
People continued to trickle into the ground throughout a lively first half.
Injury to Seamus Coleman meant that Everton lined up with Stones at right-back but with Phil Jagielka back from injury in the centre, the back four was not particularly weakened
City left playmaker David Silva on the bench and operated with Sterling behind Aguero and Jesus Navas and Kevin De Bruyne either side.
City pieced together a slick early move as Sterling played in De Bruyne but the Belgian shot wide
Yaya Toure also tested Howard with a firm header.
Aguero claimed he was tripped by Mohamed Besic when clean through but referee East gave nothing and Sterling fired over.
At the other end, Nicolas Otamendi almost turned the ball into his own net as he stretched to cut out a Gerard Deulofeu cross while Leon Osman went close to a spectacular opener as he volleyed a Romelu Lukaku cross narrowly wide.
City started the second half brighter but Everton defended doggedly and frustrated the hosts
Sterling failed to make firm contact in front of goal, Aguero hit the side-netting and Toure had an effort deflected over.
Aguero did meet a De Bruyne corner with a firm shot but Howard was in the right position to palm away and the American then stretched out a boot to deny Sterling.
Howard did even better to spread himself and block from Aguero again after the Argentinian had shrugged off countryman Ramiro Funes Mori, although the City man felt he had a penalty claim against the defender.
City sent on Silva for the final 16 minutes in the hope of prising an opening but the visitors survived as Navas headed wide and Aguero wriggled through but missed the target.
There was late drama as Aguero went down trying to beat two defenders but his appeal seemed optimistic
Sterling's penalty claim seemed stronger, and fans thought East had pointed to the spot, but instead it was merely a goal-kick.
Source : PA