"Rickie will just get better and better as he becomes more familiar with the group and becomes accustomed to the playing style," Rodgers told Liverpool's website. "I thought he was excellent and worked hard.
"I think he'll get goals for us and I'm delighted he's here. He's a wonderful character, he's 32 years of age but is still fit and strong, and he'll make a great contribution for us this year."
Lambert struck 28 goals in Southampton's previous two Premier League campaigns.