Johnson bravely headed home five minutes from time at Anfield to secure a first Premier League win in five matches for Rodgers' men.
Liverpool had looked on course for a frustrating goalless draw, but an improved second-half showing was rewarded when Johnson nodded in the loose ball after Rickie Lambert's header stuck the crossbar.
And Rodgers - whose future had been the subject of discussion following last weekend's 3-1 defeat to Crystal Palace - was full of praise for the perseverance his side showed.
"I felt we needed to up the speed and move the ball a bit quicker, in that first half we had enough of the ball but just lacked a bit of intensity, which is understandable given the confidence of the team isn't at the top yet," he said.
"The players deserve a huge amount of credit, the courage they showed today to keep going and show that resilience against a side with good players.
"[It was] a hard-fought victory for us but a well-deserved one.
"If you watch Glen, he's on the move and shows great courage and bravery - that's something the players have had.
"It's disappointing that the players haven't shown that of late, I think obviously with confidence a little bit low.
"What you have to do is show resilience and courage, then the football will come out and in spells today our football was excellent."
Rodgers named captain Steven Gerrard on the bench for the clash, which came 16 years to the day since he made his first-team debut.
However, the Northern Irishman explained the decision was based purely on the amount of football the 34-year-old has played this week.
"Steven's at the bit of his career where he's not going to play every minute of every game," Rodgers continued.
"He's obviously played against Crystal Palace and Ludogorets [in Wednesday's 2-2 UEFA Champions League draw] and we've got other players who can come in and do the job.
"Steven understands that. I wasn't actually aware of the 16 years until I walked into the changing room and saw the programme!
"He's very much a team player and a great guy - he's always plays a part in our work."
