The Reds were edged out of the competition on away goals by the Russian side despite winning the second leg of their last-32 tie 3-1 at Anfield.
Liverpool went into the game 2-0 down on aggregate and fell further behind early on but, backed by a typically vociferous home crowd, produced a rousing response to level the overall score.
Allen, who scored Liverpool's second goal between two Luis Suarez free-kicks, thinks there is plenty to be pleased about despite the performance not quite being good enough.
The Wales international said: "The intensity of our performance and the will and determination to try everything we could to get through was evident.
"There are a lot of positives. It is hard to look at them at this moment but when we look back I am sure we will.
"When you win the second leg 3-1 and still don't get through, it is pretty hard to take but I think the scoreline showed how much effort we put into the game.
"On another day we might have done enough to go through."
Liverpool are seventh in the Premier League and the result removes their last hope of silverware this season.
With just 11 games remaining some thoughts could wander to the summer and beyond, into manager Brendan Rodgers' second season in charge, but Allen still thinks there is much to achieve this term.
The 22-year-old, signed from Swansea for £15million last summer, said: "We will approach every game between now and the end of the season trying to get as many points as we can and get as high a position as we can.
"There might not be a trophy to play for but there is plenty for us as a football club to play for. We will do that."
Liverpool did have aims of Champions League football but their hopes of returning to that competition for the first time since 2009 now appear slim.
They are nine points behind fourth-placed Tottenham and could slip further back over the weekend as they are not in action.
When asked what the goals for the remainder of the season were, goalkeeper Jose Reina said: "To win - that is the next and only target.
"We need to go step by step, climb the table and make sure we win as many games as possible.
"We can only focus on the next game, which is Wigan."
Reina is disappointed by the European exit but hopes it can at least aid league form.
He said: "When you are in both competitions at the same time somehow you play better, but hopefully it will help the team and we will at least be fresher."