The 39-year-old is delighted at the club's commitment to his long-term vision, but understands that managers must be quickly successful following Roberto Di Matteo's sacking
The Reds made a stuttering start to their Premier League campaign, and were made to wait until their sixth game to record their first victory - winning 5-2 away at Norwich.
However, they are now unbeaten in their last seven league matches and with Luis Suarez finding superb form, they look to be climbing up the table and now find themselves in 11th place, four points off the top six.
"If you believe in a manager and put him in place and give him time what has been proven in this country is that can lead to success," Rodgers told reporters.
"I think those early months were always going to be difficult; you are trying to instill and implement ideas initially without a great deal of success.
"Those early months could have been difficult but in terms of myself, players and staff we were very much focused on our job and there has been no change in ourselves, only improvement.
"All you can do is have an inherent belief in how you work and I believe this is the way the club wants us to work."
Roberto Di Matteo was axed from the Chelsea job on Wednesday, after the Blues' defeat to Juventus left them on the verge of an exit from the Champions League and Rodgers suggested that, although long-term stability is important, a manager must also be successful in the short term.
The 39-year-old added: "As a manager, in order to fulfil what you want to over the longer term you have to take care of the short term - that was something I learned in my job at Reading.
"I was given a three-year contract and took the job in the great belief I was brought in to change the philosophy and ideas of the club and 20 games later I was out of a job.
"That was the harsh reality of managing a football club and from that I learned a lesson.
"I have been grateful to the board here because in order for me to come here that [support] was something which was going to be important because if we were going to keep the club moving in the right direction it was going to need some stability."