Having taken just four points from five Group B matches, the Reds needed to beat Basel at Anfield on Tuesday to finish above the Swiss side and make the last 16.
But they could only manage a 1-1 draw as Lazar Markovic was sent off on the hour mark and Steven Gerrard cancelled out Fabian Frei???s opener with an excellent late free-kick.
“The expectation was to get through the group and we haven’t done that,” said Rodgers. “It wasn’t good enough over the course of the group, but we will learn from the experience. We need to be better.
“First half we were very disappointing, our possession and our position in the game was disappointing. Technically we were way short, but second half the players were magnificent.”
Unfortunately for Liverpool, their efforts to get back into the contest were hampered by a red card to half-time substitute Markovic after 61 minutes.
The Serbian barely caught Behrang Safari with his outstretched hand but referee Bjorn Kuipers decided there was enough in it to warrant a straight red card, much to the disappointment of Rodgers.
“It was never a sending off,” he said. “If Lazar or any other player gets sent off for that, you are going to have many more throughout the competition.
“It was a disappointing decision and if anything, the player who goes down should be looked at, the fingernail barely touched his nose and he has gone down holding his face.”
Despite being down 10 men, Liverpool almost snatched victory as Jordan Henderson twice went close in the dying stages.
“The players gave everything in the second half,” said Rodgers. “The quality, tempo and speed of the game was better and it looked like they (Basel) were hanging on by the end.
“We are disappointed to go out but we showed the character to come back and give ourselves a fighting chance. The Europa League is still a prestigious tournament; there are great teams in there and it???s great experience for the younger players. It???s still European football and we will try hard to do well in that.”