Christian Gimenez scored the second for Basel on 22 minutes and Thimothee Atouba taps in the third.
But Gerard Houllier's men came out for the second half with more determination and pulled one back through Danny Murphy on 61 minutes.
Vladimir Smicer made it two three minutes later and the home side suddenly looked vulnerable.
Liverpool were awarded a penalty in the last five minutes of the game and Michael Owen fortunately netted their third, after responding fastest to his saved spot kick.
Despite intense pressure right up to the final whistle, Liverpool were unable to score the fourth goal that would have earned them a place in the next round of the competition, and instead will battle for the UEFA Cup.