The 1-1 draw was Liverpool's sixth successive league game without defeat, although it was also their sixth stalemate from 11 games, leaving them 15 points behind leaders Manchester United in 13th spot.
They are also eight points adrift of fourth place, made all the more galling by the fact that England's final Champions League qualifying place is occupied by city rivals Everton.
Three of Liverpool's next four league games are against Wigan, Swansea and Southampton, the kind of opponents they cannot afford to drop points against if they are to haul themselves up the table and into contention for a Champions League spot.
Rodgers has admitted his side were "a long way off in the league" but full-back Johnson had no doubt they were heading in the right direction.
"Of course we want to be winning games," said the 28-year-old. "Everyone can see we are starting to improve, and if we keep playing well, I'm sure the results will come our way."
Sunday's match highlighted once again just how reliant Liverpool have become on Luis Suarez, who nodded in his 11th goal of the season.
The Reds failed to muster a single shot on target before the Uruguayan equalised, and he almost snatched all three points when Petr Cech came rushing off his line to thwart him.
Johnson added: "If we had been offered a point before the game we probably would have taken it.
"A lot of teams would have rolled over and ended up getting destroyed by four or five. But we have a great spirit and a lot of heart. We always fight to the death."
Liverpool, meanwhile, are sweating on the fitness of Jonjo Shelvey for Saturday's visit of Wigan after the midfielder missed the Chelsea game with a thigh injury.