Bale scored one and set up another within 16 minutes before Liverpool launched a gallant comeback.
Steven Gerrard was denied a penalty and Luis Suarez had an effort cleared off the line before a Bale own-goal made it a tense final 20 minutes for the home team.
Andre Villas Boas‘ side hung on, and the result leaves Liverpool 10 points away from the top four.
Rodgers named an unchanged side from that which drew 0-0 at Swansea on Sunday, and the Reds were soon to find themselves behind. Bale had already tested Pepe Reina with a vicious free-kick when, in the seventh minute, the Welshman whizzed beyond Gerrard, Glen Johnson and Joe Allen before directing a perfect cross towards an unmarked Aaron Lennon at the back post.
Liverpool seemed to respond well and Suarez was soon to be denied by Hugo Lloris from 18 yards. Jordan Henderson then missed an open goal from just outside the box — though he was under heavy pressure from Kyle Walker.
These are the moments which turn games and sadly for the Reds, they were soon to find themselves 2-0 down, again thanks to the brilliance of Bale. His eighth goal of the campaign came from a lashed free-kick in the 16th minute which appeared to take a deflection off the wall and totally wrong-footed Reina.
A dangerous Stewart Downing cross was missed by both Jose Enrique and Suarez, while Enrique had a snapshot on the turn saved after excellent work from Johnson.
As the first half progressed Liverpool began to take the initiative and was denied a blatant penalty after Sandro nudged Gerrard to the ground. Suarez had a shot cleared off the line in the same incident.
Liverpool applied heavy pressure once more as the second period began, but a series of crosses and corners did not find a man.
Jonjo Shelvey was introduced for Henderson and he was to have a hand as Liverpool finally pulled a goal back after 72 minutes. It was his corner that was kept in play by Daniel Agger and Gerrard before an attempted clearance from Lennon ricocheted off Bale and into the net.
Suarez came close to an equalizer with a volley just over the bar after an overhead kick cross by Agger.
Liverpoolfc.com journalists’ man of the match: Jose Enrique.