The Robins lost in a penalty shootout after a dramatic fourth round replay at Premier League side Middlesbrough on Saturday.
They face Blackpool on Saturday desperate to rescue their promotion challenge - which has taken a hit after they managed just one point from their last three league games.
"I hope our performance at Boro encourages more fans to come down to Ashton Gate and get behind the team as we approach what is the most important part of the season," said Johnson
"Our supporters have been magnificent all season, especially away from home, but we need all the encouragement we can get and the lads deserve it.
"As they showed against Boro, they keep battling right up to the end."