The club announced on Monday that the 62-year-old Chilean will make way for Pep Guardiola in the summer regardless of what happens during the the remaining months of a campaign which could yet end with an unprecedented quadruple.
City maintained their challenge for the Barclays Premier League title with a hard-fought 1-0 victory at Sunderland on Tuesday evening, and afterwards Hart signalled the squad's intention to ensure Pellegrini goes out with a bang.
He told cityTV: "We are a professional group. We really enjoy playing under Manuel. He has been a top manager for us
"His man-management has been something that I personally have really enjoyed, so we are going to look to give him the best send-off, what he deserves.
"The fans think the world of him, the players think the world of him. We have got however many games left in the league, we are in all competitions.
"Next season will take care of itself. Right now, we are living in the moment. Manuel Pellegrini is our manager and we are going to try to win as many competitions as we can with him.
"We know if we win our games, then we will win this league. We are three points behind Leicester with not a bad goal difference compared to them, so Arsenal dropping points (in a goalless draw against Southampton) helps.
"There's a long way to go, there's all sorts left to happen. We are in a lot of competitions, we need people coming back fit, which is happening, so we are going to be in a good place."
Source: PA