Rodgers led the Reds to seventh place in the Premier League last season, a slight improvement on their previous campaign but still a disappointing finish for a club of Liverpool's calibre.
However, Liverpool's form towards the end of the season showed significant improvement, and Gerrard is hopeful that they can continue to avoid lacklustre performances throughout the upcoming campaign.
"When you get a new manager and new staff it takes a while for the message to come across and for all the players to take it on board," he told the club's official website. "But we've also got to be men about it and take responsibility that, at times, we didn't turn up and perform. There were certain games where we weren't happy with ourselves individually and as a team - Southampton away, Aston Villa at home, West Brom away.
"We have to make sure those mistakes and performances don't happen next year and, if they don't, hopefully we can have a good season. I think the manager has done well - he's taken a huge job on at a young age. He did fantastically well at Swansea and some of his other previous jobs, so he certainly deserves a crack at a big club.
"I think he's moving the team and the club forward and making good progress. If he gets the backing from the owners, and the current squad that we've already got improves, I'm sure we can make the fans happy."
Liverpool will begin the new Premier League season with a home match against Stoke City before an away trip to face Aston Villa.