The 32-year-old has scored just once in 13 Premier League appearances so far this term and has struggled for consistency.
But Rodgers feels the weight of responsibility is currently lying solely on the shoulders of his skipper, and believes the other players in his squad need to do more.
He told reporters: "Steven was outstanding last night [against Tottenham]. He's carried this club for nearly 13 years and it's about time that there was more than Steven picking up the baton of responsibility.
"He's been brilliant since I arrived here. His inspiration and leadership within the team has been phenomenal.
"He might not have got the goals that he wanted this season, but you saw him last night - his range of passing, commitment and drive into the box was superb.
"He has been judged on his goals because he's been a goalscoring midfielder over the years, but he's still offering this group so much on and off the field.
"There are others that we want to step up to the plate and get the goals. Across the board we want to improve and find the solutions."
Liverpool were denied two penalty claims in their 2-1 defeat to Tottenham and, though Rodgers refused to blame officials, he feels his side may need their luck to change.
"The referees are doing an honest job. They turn up for every game and make an honest assessment," continued the 39-year-old.
"Hopefully we can get one or two bits of luck in the second half of the season and benefit from that. I respect that they come in and do a good job - it just seems that this season quite a lot has gone against us.
"You need a wee bit of luck along the way. We went to Tottenham and dominated the game, but we didn't get the result and that's what's important.
"Over the next month or so we've got to turn these performances into wins. We had a run of eight games where we were unbeaten, looked solid and the quality of our game was good."