Rodgers was last month unveiled as the man chosen by Reds owners Fenway Sports Group (FSG) to take the Anfield outfit forward, succeeding the sacked Kenny Dalglish, under whom the team had finished the 2011-12 season in eighth place in the Barclays Premier League.
Liverpool also won the Carling Cup and were beaten FA Cup finalists last term, and despite their lowly final league position, Barnes feels the club were making progress with Dalglish at the helm.
That is something he is confident will continue next season with the new boss in charge, and Barnes is calling on Reds supporters to give Rodgers their full backing.
Asked what he thought should be expected of Liverpool under Rodgers in the forthcoming campaign, Barnes told Press Association Sport: "Improvement - we will finish better than eighth.
"If you look at the improvement Liverpool showed in the last months of the season - getting to the cup finals and the desire and resolve they showed on the field - I think there will be a natural progression of them improving, which would have happened either way.
"Brendan Rodgers is there now and I urge all the fans to get behind him, support him, believe in him and see where it takes us.
"I don't think we are necessarily strong enough to win the league but we should be looking to finish between fourth and sixth, and if we could get into fourth spot, that would be a great achievement."
Rodgers won plaudits for the work he did in his previous job as manager at Swansea, who in the two years of his tenure gained promotion to the Premier League for the first time and then secured an impressive 11th-placed top-flight finish last term - playing an eye-catching brand of flowing football all the while.
Tasked by FSG to bring the glory days back to Liverpool by employing a similar style, the 39-year-old Northern Irishman is yet to begin any major remodelling at Anfield and has made no signings yet.
Rodgers had appeared set to recruit Hoffenheim midfielder Gylfi Sigurdsson - who flourished during a loan spell under him at Swansea last season - for the Reds, but the Iceland international has since joined Tottenham.
Liverpool fans might be crossing their fingers for some big-name arrivals, but as far as Barnes is concerned, whoever the new manager does bring in, the most important thing is that they fit in with Rodgers' footballing vision.
"We have to be realistic and we have to trust the manager - know his philosophy and know he will bring in the players to suit that philosophy," Barnes, who played for the Merseyside club between 1987 and 1997, said.
"They might not be names we have heard of, like Gylfi Sigurdsson when he came in (at Swansea).
"No-one thought anything much of him, but you can see what he did.
"He is not coming to Liverpool, but if he (Rodgers) signs a player we have not heard of that he knows can fit into his philosophy and pattern, then we should support that, rather than saying 'unless we get big-name signings, we are not going to be optimistic'."
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Barnes was speaking in central Manchester at an event to celebrate the opening of the new National Football Museum.
Previously based in Preston, the museum has moved sites to Manchester's Urbis building and officially opens its doors to the public tomorrow.
It features a vast range of football memorabilia as well as a Hall of Fame, which former England winger Barnes has been a member of since 2005.