Rodgers said Suarez, who scored 31 goals last season for Liverpool but moved to Barcelona last month, had sent him a text message before Sunday's 2-1 home win against Southampton, wishing the team all the best.
But the Northern Irishman believes the sale of Suarez, for a reported fee of 81 million euros ($111 million), should not hang over the club into the new campaign.
"Our ambitions here at Liverpool have to be bigger than one player. Big clubs lose players and our ambitions are bigger than Luis," said Rodgers after goals from Raheem Sterling and Daniel Sturridge had given his team a winning start to the season.
"He sent us a lovely text this morning wishing us all the best, which was a great gesture. But he is a friend now of Liverpool.
"He's a good boy, but he's gone. His heart is with Liverpool, he will always be Liverpool and it was a lovely touch.
"But we look forward, we are bigger than any one single player," he added.
Rodgers, who has been linked with several strikers since selling Suarez, repeated his desire to bring in another forward to his squad.
"I do still think we need another striker. It is something that we will look at but that will depend on the availability of one.
"So we will search for that through to the end of the window but you see the players, the quality that we have," he said.
Rodgers started with Sturridge and Sterling in attack and new signing, England forward Rickie Lambert, on the bench against his former club.
The Liverpool manager was full of praise for Sturridge, who notched 21 goals in the league last season and grabbed the 79th minute winner.
"I think for Daniel it was just a continuation, that is 36 goals in 50 games for Liverpool which I think is a record.
"Sometimes he will have quiet games but he will always look like he will score a goal and I was pleased with that type of goal, it was a poacher's one in the box, where you anticipate the knock down and he just guided it beautifully into the corner," he said.
The loss of Suarez has led some experts to predict last term's runners-up may fail to make the Premier League top four this season, but Rodgers said that was no surprise to him.
"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," he said.
"I feel we will attack the season as we did last year - we can only do our best and we will see where it takes us.
"The fluency will come, today was about getting the win and we got the win," he said.