Cole made only his second start under Rodgers at Anfield but was withdrawn at half time after failing to make an impression on the game.
After being eliminated by his former side and denied a place in the quarter-finals, Rodgers struggled to disguise his disgust with the performance of the 30-year-old.
"I can't keep playing Steven Gerrard and Luis Suarez while Joe Allen was virtually playing on his own in midfield," Rodgers said.
"Joe Cole had an opportunity, the club has invested a astronomical sum of money on a talented player and he has to seize his opportunities.
"It was too slow, it just was not what I would expect from a team I had set up to be dynamic so I think it was a difficult night for him."
Cole spent last season on loan with French club Lille but has failed to convince Rodgers of his worth despite Liverpool's extensive injury problems.