Pellegrini retained Hart as his number one despite doubt over the England international's place following two costly errors in City's 3-1 midweek defeat to Bayern Munich in the Champions League.
"It was not a simple decision to keep Joe Hart in the team but I know that he needs all of our trust - the trust of me, the players and the fans," Pellegrini told City's website (www.mcfc.co.uk.)
"I talked a lot with him after the match against Bayern. Of course he feels he didn't do well in the last game but today he showed his strong character."
England manager Roy Hodgson, who was at the Etihad Stadium to watch City's 3-1 win, has already indicated that Hart will retain the keeper's jersey for England's next two World Cup qualifiers this month, against Montenegro and Poland.
The result gave City and Pellegrini a welcome boost after a week in which they were humbled at home by European champions Bayern and surprised in a 3-2 league defeat by Aston Villa.
"Last Wednesday had a mental and physical consequence, so I'm very pleased with the character we showed today," Pellegrini said.
He added on the BBC: "We really needed this victory because at the start of the game we were five points behind Arsenal and it is important to continue wining here at home.
"Equalising immediately was important but more important was the character of the team, playing against Everton - the only unbeaten team in the league. After a defeat on Wednesday it was important for me to see how the players reacted after that defeat."
One concern for City was the departure of captain Vincent Kompany, who limped off after 33 minutes having only returned from a groin injury in September.
But Pellegrini said the club would have to wait until Sunday to assess the seriousness of Kompany's problem.