|8/1||Liverpool are 8/1 with William Hill to finish in the Premier League top four
The Reds went into the second leg of the last-32 clash 2-0 down on aggregate after defeat in Russia and needed four goals to progress after Hulk scored early on at Anfield.
Two Luis Suarez free kicks and Joe Allen's close-range strike gave Liverpool hope but they ultimately fell short - a fact which did not detract from Rodgers's assessment of his players' performance.
"I'm very proud of the club and the players tonight," he told reporters.
"To come back from a three-goal deficit and put in that effort and quality and character into the game was just an absolutely phenomenal show.
"It was a fantastic night but unfortunately we probably lost the tie over there [in Russia]. At this level to have to score that many goals and for us to nearly do it, it gives me great pride."
The Northern Irishman went on to lament the referee's failure to award the hosts a penalty after Tomas Hubocan appeared to handle in the area in the second half.
"It’s probably too soon after the game but it certainly was a penalty, I think everybody could see that," he continued.
"We had momentum throughout the game and it looked a clear penalty, it wasn't one that there was any debate on - especially when you've got a fourth official, or fifth official standing behind the goal and looking clear at it."
Rodgers refused, however, to blame Jamie Carragher for his side's cup exit, after the veteran defender's weak backpass allowed Hulk to score the vital away goal.
"There's no blame. It's probably the first mistake I've seen him make since I've been at the football club and that includes every day in training," he added.