Benitez was jeered by supporters angry at the sacking of Roberto Di Matteo before the 0-0 draw with Manchester City on Sunday, the Spaniard's first match in charge.
"Clearly I understand the feeling of the fans, I know how they feel at the moment because of my time at Liverpool, we had a rivalry and it was quite difficult," Benitez told a news conference on Tuesday ahead of Chelsea's next game against Fulham.
"At the same time I will try to prove to them that I am here for the right reasons, I will try to prove to them I want to win, and if I can do it, they will be pleased," the former Liverpool manager added.
"I will concentrate in my coaching, try to improve players, then it will be easier to win games and bring the fans back."
Di Matteo led Chelsea to the Champions League trophy for the first time and the FA Cup after taking charge in March.
Some fans have called for Benitez to apologise for comments he made while in charge of Liverpool that he would never manage Chelsea.
"We can analyse what I say, what I didn't say, we cannot be checking the press, if it's true, if it's not true, but for me, from my point of view, is to show them that I will try my best, that I will try hard to win their trust," Benitez said.
Chelsea supporters will hope Benitez's arrival improves the form of striker Fernando Torres who enjoyed the most successful period of his career in England under the guidance of his compatriot at Liverpool.
Torres, who cost Chelsea a club record 50 million pounds last year, continued to look a shadow of his former self in the dour 0-0 draw with City.
"I'll try to give him some confidence, see what he can do in a different way and improve him - along with the rest of the team - physically," Benitez said.
"If you are strong in defence, you can regain the ball easier, you will have more chances as you will have more possession, so that's the idea. We will try to improve the shape of the team but keep their quality and mobility.
"He's not exactly the same now because he was at the maximum level at that time. He can come back and improve but the team has to help him."
Chelsea defender John Terry, midfielder Frank Lampard and striker Daniel Sturridge will miss Wednesday's match against Fulham due to injury.
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