Dejan Lovren headed home Adam Lallana's second-half corner - defending Rodgers described as "criminal".
"I can't underestimate how well the players have done," he said.
"This is our first defeat since March, funnily enough against Southampton.
"We've had two poor performances against them and it's cost us.
"We are very disappointed. We never got going from the off, it was a bit lethargic and technically we were short."