The Reds trailed 2-0 from the first leg and after conceding an early goal at Anfield it seemed the tie was all wrapped up, even by the standards of a team who came back from three goals down to beat AC Milan and win the Champions League in 2005.
A double by Luis Suarez and a Joe Allen strike gave the hosts half an hour to get that crucial fourth, but it was all to no avail as a 3-1 victory was not enough to prevent Liverpool from going out on away goals.
"I thought we were incredible," Rodgers said. "Obviously we were two down from the first leg and to concede again, at 3-0 you might have thought the tie was over and done with.
"But we showed real character and quality tonight to get three goals, and we totally dominated the game, controlled the game and should have had a penalty.
"So we're bitterly disappointed and we lost the battle tonight, but in terms of the war going forward we very much look forward to that. Overall we were fantastic.
"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."
Rodgers refused to criticise Jamie Carragher, whose 150th European appearance for the club turned out to be his last given his recent decision to retire at the end of the season.
The 35-year-old was at fault for the opening goal when he miscontrolled the ball and Brazilian striker Hulk stole in to score.
Rodgers added: "There is no blame. It is probably the first mistake I have seen him make since I came to the football club, and that includes every game in training.
"I thought he was brilliant tonight, how he controlled Hulk, who is a real threat with his strength and power. It is one of those unfortunate things.
"I am disappointed for Carra. He is a brilliant professional and deserved to go into the next round and play again, but unfortunately it is not to be."