Emile Heskey had put Liverpool ahead in the first half but Fulham equalised through Louis Saha and it looked like the points would be shared.
But Pongolle, on for Michael Owen for the last 15 minutes, injected pace into the Liverpool attack and rightly earned a penalty in the dying minutes.
Murphy kept his cool and slotted home his second penalty of the week to seal the victory.
Who was Liverpool's Man of the Match?