The game was short on drama and chances with the main talking point being Sylvain Distin's disallowed headed goal in the second half.
Aside from that, the best, and only real genuine chances fell to Toffees' midfielder Marouane Fellaini who volleyed wide from four yards and Liverpool captain Steven Gerrard, who rounded goalkeeper Tim Howard only for Distin to clear his goalbound shot.
"We weren't at our best but what you need to make sure is you are competitive," said Rodgers.
"We kept another clean sheet, which is 15 (in the league) this season and we need to be strong defensively as well as attacking.
"We've scored a lot of goals this season but we obviously need to improve in that aspect as well to break down teams who come and play that bit deeper.
"We need to find one or two players who can make us help the difference in games like this.
"We've had a lot of draws this season and we need to find the solution for next year."