The unlucky Tim Howard then diverted in a third in the 69th minute after initially saving Aguero's penalty.
The victory was only City's third in 13 attempts against Everton, who tasted defeat for the first time in the league under Roberto Martinez.
Despite his refusal to confirm that Hart would start following his mistakes against Bayern, City boss Manuel Pellegrini showed faith in the England man while changing more than half his team.
Only Vincent Kompany survived from the defence that faced the European champions while David Silva, James Milner and Negredo were also included.
Everton boss Martinez was without on-loan Gareth Barry, ineligible against his parent club, and the Toffees missed the midfielder's calm presence in a frenetic first half.
Hart was straight into the action, diving on a ball from Steven Naismith with Lukaku lurking, but it was City who had the best of the early chances.
In the fifth minute, Howard pushed a shot from Yaya Toure behind, while two minutes later half the crowd thought the hosts had taken the lead when Aleksandar Kolarov's free-kick flew into the side-netting.