Gerrard also speaks of how each morning during the summer he hoped Suarez was still a Liverpool player
Brendan Rodgers’ side moved up to second place and just two points behind leaders Arsenal with a 3-1 win at rock-bottom Sunderland on Sunday.
It ended a mini-slump with Liverpool having not won in three games and suffered successive defeats to Southampton in the Premier League and Manchester United in the Capital One Cup.
The Anfield outfit’s total of 13 points from their opening six matches is their best start since the 2008-09 season, when they finished second.
And Gerrard believes that with their near rivals all struggling for consistent form, Liverpool must remain among the front-runners going into the festive period.
“Everyone is taking points off one another and I’m sure other teams are taking note that we are a threat to them,” said the Liverpool skipper.
“But to be honest, I’m not too bothered about what people are saying about us. I know what this team is capable of.
“The main thing for us is to keep fighting and show that we are capable of staying in these positions to around Christmas time and then we will see where that takes us. The only way we will do that is by winning games.
“It’s very important that we get ourselves into a good position because there is nothing worse than going into the New Year having left yourself too much to do.”
Suarez marked his return to Premier League action following his 10-game ban by scoring twice at the Stadium of Light.
The Uruguayan attempted to force through a move to Arsenal during the summer but backed down when it became clear Liverpool were unwilling to sell.
Gerrard is convinced the 26-year-old is now once again fully committed to the club, but when asked if he was worried the striker may leave he responded: “Every day in the summer.
“When you have a player as good as Luis Suarez, you know that the big clubs are going to want him.
“I was waking up every morning, hoping that nothing would happen and that he would still be here.
“Nobody tried harder than me to do that (make him stay) and he knows exactly what I think of him. I knew if we could keep him here, the better our chances would be.
“He is a special type of footballer and he gives us something different. When the club got the message to him that they were not going to sell him, he got his head down and has worked hard.
“He has never given us a moment’s problem.”
Gerrard added: “I think it is a credit to Luis for the way he has come back. A lot of players who have been in his position might have come back and spat the dummy out, not trying for the team.
“But he is not that type of guy. He realised that the club wanted to keep him and he has respected their decision.
“He is here now and he wants to stay. He wants to achieve something this season and he looks as hungry as he has ever done.”
Both of Suarez’s goal were assisted by Daniel Sturridge, who himself netted to take his tally to seven for the season and 11 in his last 11 outings.
And Gerrard believes such form can see the forward leading the line for England for the forthcoming decisive World Cup double header against Montenegro and Poland at Wembley.
“For me, Wayne Rooney and Daniel Sturridge is the partnership that every England fan wants to see,” said the England captain.
“You have seen what he offers and if we can keep Daniel and Wayne on the pitch, so much the better. He is in top form and has made a brilliant start to the season.”