Rodgers' side took an early lead through Adam Lallana, but they were pegged back by Ebenezer Assifuah's first-half equaliser.
The Reds wasted a host of opportunities to regain the advantage and under-fire boss Rodgers was forced to endure boos from the Kop at the final whistle.
With a Merseyside derby at Everton looming on Sunday, Rodgers knows the scrutiny will only grow in the coming days, but he maintains he is still relishing the task of getting Liverpool back on track.
"We are one point behind Everton. I know the media and written press will build the game up but we will stay calm," Rodgers said.
"I am immensely enjoying the job, it is a great challenge and it is a rebuilding job again."
Liverpool have won only one of their last eight matches in all competitions, increasing the pressure on Rodgers.
But he still made seven changes as he rested key players ahead of the Merseyside derby.
And the replacements, especially Belgian forward Divock Origi, who was deputising for Daniel Sturridge, failed to take their chances.
"This is where we find out about players like Divock Origi and others on the fringe," Rodgers said.
"They will be important for us in games like these. Divock did well but he didn't score and he has to add that to his game.
"We go into the game against Everton and we will focus on our performance. The players will look forward to it. I haven't lost a derby yet and they are always good games."
But Rodgers did concede he was unhappy with the way his team once again failed to kill off inferior opposition.
"The result was disappointing considering the number of chances we created," Rodgers said.
"We worked well and had a good start but gave away a disappointing goal.
"The players worked hard but we couldn't score the goals. You have to be clinical. We scored a good first goal.
"They weren't half chances, they were good chances and you have to score them at this level. We have to win a game like that."
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