The Reds drew a blank at Swansea on Sunday as manager Brendan Rodgers returned to the club he left for Anfield in May.
Rodgers will reflect on a contest his side probably should have won after having Jose Enrique's strike ruled out for offside, seeing Raheem Sterling hit the bar and Suarez force two sharp saves from Gerhard Tremmel.
But the inability to find a breakthrough adds further weight to the impression that if Suarez does not find the net, neither do Liverpool.
Suarez has scored 10 Premier League goals so far this season; no other Liverpool player has managed more than one.
But Johnson, who excelled at the Liberty Stadium, does not feel the Merseysiders are too dependent on their £22m front man and echoed his manager's belief that Enrique's first-half effort should have stood.
"We are not relying on Suarez at all," Johnson said. "Obviously he is our main striker, but we are not relying on him.
"It's everyone's job to get around him and try and score.
"You need a bit of luck in front of goal and on another day we might have had two or three. It's a difficult one because with Jose's goal I thought his feet were in line with the last defender.
"His shoulders were offside but surely the advantage should go with the attacker. I have seen them given. It is frustrating for us but you can see why the linesman has given it if he can see some part of his body in front. I never felt that we were in any danger but they dug in and defended well."
Liverpool are now eight games unbeaten in the top flight having recorded four wins and four draws.
Sunday's 0-0 draw leaves them 11th in the table, still a point and two places below Swansea, but Johnson believes the Anfield club now have a solid base from which to climb into the upper reaches of the standings as they prepare to face Tottenham on Wednesday.
"It is a good platform. If you can't win a game you've got to make sure you don't lose it," he said.
"We are sitting there and haven't had that much luck. We are not relying on luck but it's a day like this that you could go home with a 1-0 or 2-0. But we will keep going and if we do get that bit of luck it will be okay.
"We will recover from this and turn our attention to Spurs. It is a big game and we have a couple of days to prepare for it and hopefully we will pick up three points."