The 21-year-old Italian was the new manager's first signing in a ?10million deal from Roma last month.
Rodgers knows the youngster well having coached him at Chelsea and also had him on loan at former club Swansea.
The player got his Liverpool career off to a good start with a goal on only his second appearance, scoring the opening goal in the 3-0 Europa League qualifying win over FC Gomel.
And while he is far from the finished article Borini will have to step up a level this season in a forward line in need of a regular source of goals with Luis Suarez and the out-of-favour Andy Carroll the only recognised strikers.
"Expectancy is what comes from being at a big club so you have to deal with it both as players and manager," said Rodgers, who is looking for his side to bounce back from their opening-day defeat at West Brom at home to champions Manchester City on Sunday.
"He is not alone. When you come to a big club there is always big expectation and pressure.
"This is a kid who works well and works hard and he will come good there is no question about that.
"The most important thing is he is fighting for the shirt.
"The biggest thing in football you need is perseverance. All these players are persevering and want to work."
Rodgers does not have the funds to bring in a much-needed top-class goalscorer - although he is on the verge of securing a loan deal for Real Madrid midfielder Nuri Sahin - but insists that will not necessarily be the cure to their problems which stem from missing a host of chances last season.
"It is no good having a goalscorer scoring 30 goals and no one else scores - I've always been against that theory," said the Reds boss.
"You are going to need upwards of 70-odd goals to be up around the Champions League [places].
"For me it is a team ethic. Luis Suarez will score goals, Steven Gerrard can score goals, young Borini will score goals.
"But it is a team responsibility, not just one player. You bring in one striker who can score you goals, which is great, but it's not good for the team if the rest don't score.
"For us it is trying to have players in different areas of the field who can cause a problem.
"I told the players this year 'You can't blame referees and crossbars, you have to be clinical, you have to be ruthless'.
"But equally important is that at the other end you are not conceding goals.
"I think we have shown, apart from last week (in the 3-0 defeat at West Brom) when the game was broken when we went down to 10 men, that we have a good mindset and be strong this year."