The Merseyside club came agonisingly close to a first league title since 1990 in May, but were just pipped by Manchester City.
Following the sale of star striker Luis Suarez to Barcelona, Liverpool have struggled to find form in the early weeks of 2014-15 and have lost three of their first five Premier League fixtures.
Tuesday's third-round tie against Middlesbrough at Anfield represents a chance for Rodgers' men to pick up a much-needed win and regain some confidence, with the Northern Irishman searching for an improved display from the one seen in Saturday's 3-1 reverse at West Ham.
"We've made mistakes in games and we need to cut those out," he said in a press conference on Monday.
"We just have to go back to working and moving the ball but on top of that we need to be more aggressive with our defending.
"As a team we need to be better. We started poorly at the weekend and you see [in] the results [that] there is no easy game at this level.
"You will be tested physically and mentally at this level and you have to be ready to fight that.
"It is a long campaign and we have a lot of games. We haven't made a great start by any means but we'll improve.
"We need to concentrate on the game [against Middlesbrough] because we need to get through.
"We want to win a trophy this season, I've said that before and this competition is an opportunity to do that."
Rodgers confirmed that Rickie Lambert will make his first competitive start since joining from Southampton in the close-season, but Jordan Henderson (hip) and Philippe Coutinho (groin) are unlikely to feature, with the Liverpool manager set to give some of the club's academy players a chance to shine.
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