The Reds have seen their dreams of winning a final European trophy under Jurgen Klopp dashed.
After trailing Atalanta 3-0 from the home leg, the Reds could only manage a 1-0 win in Italy.
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Speaking to TNT Sports afterwards, Van Dijk stated: Firstly, credit to Atalanta.
The damage was done last week (when) we were very poor and they had the perfect game. They deserve to go through. We made it far too difficult for ourselves with the result last week.
Today was a good game, great fight. Obviously we won but it doesn't feel like it really. Obviously it's an improvement, but the reality is that we are out and we have to quickly turn this around, recover and travel to London on Saturday.
We need everyone for that last push, and that's the feeling that I'm getting now."