Jürgen Klopp's side were in control for much of the first half but fell behind to an Olivier Giroud header shortly after the half-hour mark.
And the Reds could not find an equaliser in the second period, meaning they will likely have to beat Brighton and Hove Albion next Sunday to secure Champions League qualification.
See the story of the game in photographs below.
Images brought to you by Western Union, official principal partner of Liverpool FC