Steven Gerrard's 81st-minute free-kick earned Liverpool a 1-1 draw with the Swiss champions at a drizzly Anfield on Tuesday, but it was the visitors who progressed to the last 16 alongside group winners Real Madrid.
Liverpool were making their return to the competition after a five-year absence, and with six points separating them from the Premier League's top four ahead of Sunday's trip to Manchester United, Rodgers admitted his side must improve if they are to earn a place in next season's competition.
"I think we earned the right to be in this year after a wonderful season last year, but over the course of the games we weren't good enough and ultimately went out," he told his post-match press conference.
"But it's been a really good learning experience for us and we want to fight as hard as we can to make sure it's not another five years. We are bitterly disappointed and we will fight very hard to make sure we are back in it next year."
Meanwhile, Basel coach Paulo Sousa said his side deservedly held on. “We all made history for this marvellous club FC Basel today," he said. "My team has fought and we had fantastic support from the fans in the stadium. Now the story continues! "
Real complete perfect group stage
Real Madrid coach Carlo Ancelotti hailed his ‘extraordinary’ players after they set a Spanish record for consecutive victories of 19 thanks to a 4-0 success at home to Ludogorets in Champions League Group B on Tuesday.
Real surpassed the previous record they jointly held with Barcelona, who won 18 matches in a row in all competitions between October 2005 and January 2006. European champions Real last failed to win when they were humbled 2-1 at home by city rivals Atletico Madrid in a La Liga game in September.
"I have an extraordinary group [of players] and I never tire of saying it," said Ancelotti. "I could not imagine winning 19 games in a row after the defeat to Atletico. I knew we would fix what had not gone well but not in this way."
Wenger's critics silenced for now
Aaron Ramsey and Lukas Podolski got Arsenal back on track with a brilliant doubles in Tuesday's 4-1 win at Galatasaray, but there was still frustration for under-fire boss Arsene Wenger as the Gunners were unable to clinch top spot in Champions League Group D.
Although Wenger's side travelled to Istanbul already assured of their place in the last 16 of Europe's elite club competition, they were sorely in need of a morale-boosting victory following the vicious backlash from Saturday's 3-2 defeat at Stoke.
Finishing second means Arsenal are likely to face tougher opponents in the last 16, which has been their regular exit point from the tournament in recent seasons, so Wenger's critics are unlikely to be silenced for long by victory over Galatasaray, who got one back through Wesley Sneijder.
"There's a little bit of frustration that we have finished runners-up, but we knew at half time we needed to score six goals without conceding," said Wenger.
"Unfortunately it wasn't enough, but we finished with 13 points and did the job tonight.
"We came out strong and that's what I wanted from these players."
Meanwhile, on a freezing night in Turin, Juventus dominated last season's beaten finalists Atletico but failed to even threaten winning by the two-goal margin that would have helped snatch top spot in Group A as they played out a goalless draw.
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