Rodgers pleased by post-Suarez Liverpool
Uruguay striker Suarez departed Liverpool for Barcelona during the close season and there were question marks over how Liverpool would cope in his absence.
Liverpool were far from their best against Southampton in their opening Premier League game of the season at Anfield on Sunday.
"Our ambitions here at Liverpool have to be bigger than one player," Rodgers said.
"Luis sent us a lovely text (message) this morning wishing us all the best, which was a great gesture. He is a friend now of Liverpool. He's a great boy, but he is gone now.
"We are bigger than any player. We do still want another striker, but that will depend on the availability. It was all about winning against Southampton and we got there.
"As much as the good football we play, it was about grinding it out so I am really happy with that determination and character.
"It was very hard-fought. Just because Southampton have lost players, it's not going to be a walkover.
"The players are still searching for their fitness, they are still getting their condition and getting themselves back into it. We weren't as fluent, but it was about the win."
Sterling put Liverpool ahead with a composed finish from Jordan Henderson's sublime through-ball, but Clyne fired in Southampton's equaliser following a clever flick from debutant Dusan Tadic.
However, Liverpool secured maximum points when Southampton failed to clear a cross, allowing Sterling to head the ball into the path of Sturridge, who scruffily nudged it past Fraser Forster.
- 'Don't think we'll struggle' -
Despite a second-placed finish last season, Liverpool are not among the sides widely expected to challenge for the title in this campaign, with Chelsea and Manchester City considered more likely winners.
But Rodgers said that his squad would thrive on being written off.
"We believe in the way we work," he said. "I don't think anyone is giving us any hope of getting in the top four, but that was the expectancy last season as well.
"We love the challenge. We go into that, look to recreate that, by winning games and performing.
"We are bringing in young players who are ready to go in the team, but there is still a lot of improvement."
Southampton's new manager Ronald Koeman has had to contend with the departures of five key players during the close season.
But the Dutchman, who replaced Mauricio Pochettino at St Mary's after the Argentine left for Tottenham Hotspur, claimed Southampton showed at Anfield that they are still a competitive side.
"It is between the white lines of the pitch to show the people and first to show ourselves that we have to believe in our quality," he said.
"That was one of the messages we gave the boys. We played an impressive second half and that is how we like to play. If we play like that, I don't think we will struggle."
Koeman also said that he had no concerns over the selection of Morgan Schneiderlin -- who has been also linked with a move away from Southampton -- and will continue to pick the France midfielder.
"Morgan is a player of Southampton and will continue playing for Southampton," the former Ajax coach said.
"He changed his mind and mentality and he showed that this week. He showed his qualities and he is very strong for the team. It was normal for me to put him in the line-up."
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