The Reds are still without a win in the Premier League this season after defeats to West Bromwich Albion and Arsenal as well as a draw with Manchester City.
Rodgers took over at Anfield after a successful spell with Swansea City, and Johnson has called for the supporters to give the new boss some time to turn things around.
"It's clear for everyone to see the way Brendan wants us to play. He did fantastic at Swansea and he's a fantastic coach," said Johnson. "When Brendan is talking, you're so focused on what he's saying. He gets your attention 100%. You can be in a meeting for half an hour with him, but it'll feel like 10 minutes.
"Anything new is going to take time to mould, but the lads are working hard every day to try and make the success come together.
"Everyone needs to be patient, including the players. We need to make Anfield a place where no team wants to come or can enjoy their football. We need to get on top of teams from the start and try and make it as difficult as possible for them for the 90 minutes."
Johnson is currently on international duty with England.