Rodgers took over the five-time European Cup winners in 2012 and finished seventh with them in his maiden campaign, before jumping to second last term, narrowly missing out on their first title for 24 years.
The Northern Irishman earned widespread plaudits in the process as they were eventually pipped to the post by Manchester City.
However, Carragher, who played for the Reds 737 times, believes that the feelgood factor surrounding the club must now be furthered with some silverware.
"Next year it will be important that they do well in the Champions League and the Premier League, but also try to win a trophy as well.
"The manager has been there for two years now; the next step for him and the team, a lot of the players, is to pick a trophy up and get some winners' medals."
Rodgers has already been decisive in the transfer market as he plans for life in the Champions League next season.
The Reds officially landed Rickie Lambert from Southampton earlier this week, while a £9.75 million move for Bayer Leverkusen midfielder Emre Can is set to be finalised over the coming days.
Rodgers has also given the green light on a £18 million pursuit of Sevilla left-back Alberto Moreno.