The game between two title contenders lacked in clear-cut goal chances and was more notable for the scenes that unfolded in the stands at Stamford Bridge, as Chelsea fans welcomed Roberto Di Matteo's successor with boos.
"If we start winning games, they will come on board and they will see I'll try to do my best," Benitez said.
Chelsea has fallen to fourth after three draws and two losses in its last five league games, behind West Bromwich Albion.
While Benitez's former side Liverpool also drew 0-0 on Sunday against Swansea, Tottenham rose to seventh by beating West Ham 3-1 and Southampton climbed out of the relegation zone with a 2-0 win over Newcastle.
While the focus was on Benitez on Sunday, City did manage to maintain the unbeaten start to its title defence, although Roberto Mancini's side is a point and a place behind leader Manchester United.
And while Mancini held on to his job despite City exiting the Champions League in midweek, Di Matteo paid the price for a loss at Juventus that still left Chelsea with a chance of advancing into the knockout phase to continue its title defence.
Managerial changes, though, are the norm at Chelsea, with Roman Abramovich now having hired his ninth in his nine years as owner.
Never before, though, has the Russian billionaire seen a manager endure the kind of abuse from fans witnessed on Sunday.
"He knows it was a tough game," Benitez said after speaking to Abramovich. "He knows it was a difficult time because of the last games."
Bentiez's first introduction to the crowd was greeted with boos, which persisted during the opening stages of the match, as fans held up "Rafa Out" banners.
"My experience in England is that when the fans are singing I don't understand what they say," said Benitez, whose last job here ended at Liverpool in 2010. "I can understand the rivalry in the past (between Chelsea and Liverpool), but I am sure the majority of fans will understand I am a professional and I just want to do my job."
And there's a sizeable job to do to get the Blues reproducing the form that saw them top the league early in the season.
Fernando Torres, who was brought to Liverpool was Benitez, has lost his scoring touch since joining Chelsea for 50 million pounds (then $71 million) in January 2011. And the striker cut a peripheral figure on Sunday, with just one threatening moment squandered as he blasted the ball over in the second half.
"You could see he was trying hard — the team has to help Fernando create more chances to score goals," Benitez said.
Ramires also struck over for Chelsea, while David Luiz sent a header off target.
City's most dangerous openings were in the first half, with David Silva heading over and Sergio Aguero nodding straight at goalkeeper Petr Cech.
"We were so poor in the last 20 metres," Mancini said. "When we have a chance, we need to score. If you are soft, you won't score. That's why I was upset."
Tottenham's match against West Ham was overshadowed by more distasteful chants directed at Spurs.
"I prefer not to mar the performance with a situation like this," Tottenham manager Andre Villas-Boas said.
On the pitch, Tottenham's run of three consecutive league losses ended as Jermain Defoe scored once in each half either side of Gareth Bale's goal.
West Ham's on-loan striker Andy Carroll scored a late consolation, his first goal in 11 league appearances since moving from Liverpool in August.
On the south coast, Southampton kept a clean sheet for the first time in the league this season as goals from Adam Lallana and Gaston Ramirez lifted the team out of the drop zone, a point above Aston Villa. Newcastle is just three points ahead of Southampton.