The Reds responded to falling 3-0 behind on aggregate early in the second leg of their last-32 tie at Anfield by feeding off a vibrant crowd to level the overall score.
Luis Suarez struck twice from free-kicks and Joe Allen tucked in at the second attempt from close range but Hulk's vital away goal - gifted to him by veteran Jamie Carragher - ultimately proved decisive.
The Reds produced a host of other chances to take the lead and also had a strong penalty appeal turned down after Tomas Hubocan appeared to handle.
Rodgers said: "I'm very proud of the club and the players.
"I think, even though we lost the tie, the football club and players showed they are part of probably the world's greatest football family, here at Liverpool, with atmosphere and what the players put into the game.
"To have to come back from a three-goal deficit and put that effort and quality into the game, it was an absolutely phenomenal show.
"I don't think the players could have done any more, the crowd certainly couldn't have done any more. It was one of those nights.
"It was a fantastic night but unfortunately we probably lost the tie over there.
"At this level, to have to score that many goals and nearly do it gave me great pride."