The England goalkeeper hit the headlines for the wrong reason last week with high-profile mistakes in City's 3-1 UEFA Champions League defeat by Bayern Munich.
It is not the first time Hart's form has been questioned over the last couple of seasons, and he might have done better for Everton's goal in City's 3-1 win at the Etihad Stadium on Saturday.
Romelu Lukaku's shot was well struck and from close range, but Hart got a good hand to it only for the ball to bounce into the net anyway.
Full-back Zabaleta revealed Hart was unhappy with the goal but expects him to rediscover his best form.
"We saw him during the week a little bit down," said Zabaleta. "We need him to come back again to his best.
"Sometimes it happens in football but that's why team-mates need to support all the players. He knows we're all behind him.
"Of course he's upset. It's really hard for any player when you know you're not doing well. He has a personality so he needs to show that he's coming back to his best. And I'm sure he will do because he's a great goalkeeper.
"We've seen him for a long time doing very well so that's why it's strange to see him like this."
The home crowd were particularly vocal in support of their number one, who joined City from Shrewsbury in 2006.
Zabaleta said: "He's been here for a long time and the fans are behind him, everybody's behind him. I felt very good when I saw the crowd giving massive support to him. We need to be all together in this time."
City boss Manuel Pellegrini admitted he thought hard about picking Romanian Costel Pantilimon ahead of Hart before deciding to stick with the England keeper.
Zabaleta does not expect the competition to adversely affect Hart, saying: "Costel is another great goalkeeper and he is putting pressure on him, and that is great.
"I think for everyone, we know we've got two good players in every position and we need to feel the competition. That is normal."
After the loss to Bayern and a previous league defeat by Aston Villa, it was an important victory for City against a team who had not lost a Premier League game under Roberto Martinez until Saturday.
"We needed to win," continued Zabaleta. "I think it was only about winning the game after two defeats in a row.
"It was a difficult week for us, especially the game we lost against Bayern Munich. We needed to respond well and I think we did it.
"We conceded an early goal and it was important to score two goals in the first half. Then the second half I think we played well."
Zabaleta was the architect of the third goal when he went down in the box under pressure from Seamus Coleman and referee Jon Moss gave a penalty.
Everton were furious, and Zabaleta conceded the decision was a "soft" one.
He said: "I think there was small contact. It was maybe a silly penalty. I watched it in the dressing room and maybe it was a little bit soft but I think it was a penalty.
"He just pushed me away a little bit. I can understand that Coleman was a little bit disappointed. But I was talking with Leighton Baines after the game and after the penalty and he understood.
"And when the referee showed me a yellow card for a foul against him he went to the referee saying I touched the ball and it was no foul. I like that."