On this showing though it is the Reds who have most to fear from the drop as the Seasiders fully deserved the three points after going in at the break 2-0 ahead, conceded an early goal in the second half but then defended well to frustrate the home side.
Varney then doubled the visitors advantage with a goal on the stroke of half time. The home side were booed off as they left the field.
A 54th minute Sotirios Kyrgiakos header gave the Reds hope and he almost levelled in injury-time but was denied by a superb save by keeper Matt Gilks.
Liverpool scorers: Kyrgiakos, 53
Blackpool scorers: Adam (pen) 29, Varney 45+1
Referee: Mike Jones
Venue: Anfield, Liverpool
05 Eardley (Phillips 46)
20 Cathcart (Keinan 20)
14 Grandin (Southern 63)
12 Taylor-Fletcher (yellow card)