Benitez was quickly installed in the Stamford Bridge hotseat after Tuesday's 3-0 defeat by Juventus spelled the end of Roberto Di Matteo's reign at Chelsea.
The arrival of the Spaniard has not been warmly greeted by Chelsea fans, who remember his fiery clashes with former Blues boss Jose Mourinho during his spell at Liverpool, but Redknapp feels they will forgive and forget if he inspires instant success against City.
Chelsea have slipped to third in the Premier League table after a run of four games without a win and former Tottenham boss Redknapp knows that Benitez does not have much time to settle in ahead of the crucial top-of-the table clash with the defending champions.
He told The Sun: "I know Benitez isn't a popular choice, but the ball's in his court.
"I have always said the punters do not really care who's in charge, so long as the team is winning and the points are racking up. You could have Saddam Hussein in charge and if results went well they would be chanting his name.
"So Rafa has to start by beating City. If he does that, it'll be a different story."
Redknapp has been out of work since parting company with Spurs in the summer, but he is expected to make a flying visit to the Ukraine to hold further negotiations about taking charge of the national side.
Ukraine Football Federation chief Mykhailo Fomenko said: "Talks with them were successful, so now we're waiting for what Redknapp will say.
"In the next few days we expect Redknapp will make a personal visit to thoroughly discuss all the details and conditions of the possible contract and get his final decision."