The Reds remained in the drop zone as the Seasiders jumped to ninth thanks to a 29th minute Charlie Adam penalty and a late first-half goal from Luke Varney.
The two-goal lead was cut back by Greek defender Sotirios Kyrgiakos who headed home a Steven Gerrard free-kick eight minutes after the break but it was not enough for Roy Hodgson and his men who cannot take a trick in their dismal start to the new campaign with a 10th minute groin injury to striker Fernando Torres compounding the misery.
LIV Pepe Reina (gk), Sotirios Kyrgiakos, Martin Skrtel, Jamie Carragher, Glen Johnson, Christian Poulsen (Jovanovic 60'), Steven Gerrard, Joe Cole (Rodriguez 89'), Raul Meireles, Fernando Torres (Ngog 10'), Dirk Kuyt
Subs - Brad Jones (gk), Maxi Rodriguez (89'), David Ngog (10'), Lucas, Jay Spearing, Martin Kelly, Milan Jovanovic (60')
BLA Matthew Gilks (gk), Ian Evatt, Craig Cathcart (Keinan 21'), Stephen Crainey, Neil Eardley (Phillips 45'), Charlie Adam, David Vaughan, Eliot Grandin (Southern 64'), Luke Varney, DJ Campbell, Gary Taylor-Fletcher