Both Spanish forwards - who joined the English giants in the close-season - have failed to score in their six Premier League appearances each so far this term.
But with Sturridge sidelined for up to two months with an ankle injury, Rodgers needs players to step up and support Uruguay star Luis Suarez.
"We can't just rely on Luis Suarez for goals," Rodgers told the club's official website.
"The likes of Aspas and Alberto have been at the club for a few months now and have seen the difficulties of playing in the Premier League, how hard it is and the physical demands.
"I can see them improving and working hard to fit into it. They will get improve and get better in time. There was always going to be a period of adjustment.
"With Daniel out, other players who have been working hard will get an opportunity. We now have to find a different way to win games. We can't just rely on one player for goals. It's the responsibility of everyone.
"It has to be for everyone to create and score goals. If some aren't available, we need to do more as a team - that was the message after the defeat to Hull."
Philippe Coutinho is likely to return to Liverpool's starting line-up against Norwich City on Wednesday after overcoming an ankle strain.
