There were doubts over the 26-year-old's position after he was culpable for two of Bayern Munich's goals in City's Champions League defeat on Wednesday.
It is not the first time recently that Hart has made high-profile mistakes, and Pellegrini pointedly failed to confirm whether he would pick Hart for the clash with the Toffees in the build-up to the game.
Roy Hodgson has assured Hart he will start in the forthcoming World Cup qualifiers, and the England boss watched on from the stands at the Etihad Stadium as questions were again asked of Hart.
The City stopper got his hand to Romelu Lukaku's shot in the 16th minute but could not keep it out.
Pellegrini did not blame Hart, saying: "Always the responsibility for the goal is with the whole team."
But asked about his decision to choose Hart ahead of Costel Pantilimon, Pellegrini said: "It wasn't a simple decision, it was a very difficult decision, but I think Joe needs the trust of all of us - of me, the other players and the fans.
"Of course he feels he didn't do well the last game, he feels that he was guilty, but it was more important that he reflected how strong his character is."
This time the goal did not prove costly as Alvaro Negredo equalised almost immediately and Sergio Aguero put City ahead just before half-time.
The third arrived in the 69th minute, Tim Howard saving Aguero's penalty only for the ball to rebound off a post and back in off the Everton keeper.