The Reds went ahead when Cole set up Ryan Babel to finish with aplomb.
Ryan Babel controlled and fired home a terrific second before Steven Gerrard's shot squirmed under Giedrius Arlauskis.
And Schwarzer was again equal to Ryan Babel's fierce strike as Fulham defended deep and frustrated Liverpool.
Offers for Liverpool attacker Ryan Babel, Celtic winger Aiden McGeady and Tottenham striker Roman Pavlyuchenko were all turned down.
Ryan Babel spurned a glorious opening for the visitors when his strike was saved with only the keeper to beat.
Jamie Carragher and Ryan Babel had reasonable chances, while Torres and John Arne Riise were thwarted by some well-timed tackles.
Hodgson also praised Dutch midfielder Ryan Babel who marked his first Premier League start of the season with a goal.
But Kolo Toure fouled Ryan Babel and Gerrard held his nerve from the spot, before Babel broke to add a fourth.
The visitors were starting to look jaded, and Benitez replaced captain Gerrard with the pace of Ryan Babel on the hour.
Liverpool's wide players, Yossi Benayoun and Ryan Babel, also found the going tough - as did Blackburn's wingers Morten Gamst Pedersen and Bentley.
Ryan Babel set up Dirk Kuyt to put the Reds ahead and only some super saves by Luke Steele kept Barnsley in the game.
Yet within 30 seconds Ryan Babel jinked into the penalty area to shoot low and hard, forcing a superb stop from Luton keeper Dean Brill.
Steven Gerrard's cross flew it to make it 4-1 before Ryan Babel tapped in Dirk Kuyt's cross and then unknowingly deflected in Albert Riera's late shot.
Ryan Babel equalised with a glancing header from Lucas Leiva's cross and Albert Riera put them ahead with a 25-yard drive that found the top corner.
"We got a penalty two years ago in the quarter-final of the Champions League at Liverpool that made the difference when Ryan Babel dived, " said the Frenchman.
Fortune then had an effort disallowed for offside while Ryan Babel and Lucas Leiva both missed chances at the other end in an entertaining enounter at The Hawthorns.
"You have Maxi Rodriguez, David Ngog, Ryan Babel, Joe Cole - they are quality players and it is a very strong squad and they look organised, " he said.
Ngog also replaced Torres in the closing stages of the Europa League game with Benfica after the Spaniard was deployed as a lone striker following Ryan Babel's first-half dismissal.
Ferguson and Benitez made a single change each to their teams, with Park replacing Dimitar Berbatov for United and Maxi Rodriguez coming in for Ryan Babel in the Liverpool line-up.
The goal gave Liverpool a further injection of confidence and they dominated the early stages of the second half, with Ryan Babel fizzing a 25-yard shot just wide after 58 minutes.
Houllier admitted that conceding two goals in the opening 16 minutes at Anfield, with one each from makeshift strike-force David Ngog and Ryan Babel, made things very tough for his young side.
Some unfamiliar names are also listed such as Chris Nathaniel, a young agent who has negotiated a seven-figure book deal for England's John Terry and works closely with Ferdinand, Micah Richards, Obafemi Martins and Ryan Babel.
Liverpool's three-pronged attack of Ryan Babel, Dirk Kuyt and Torres were moving the ball around crisply and a second goal almost arrived after a Gerrard corner, but keeper Mickael Landreau was alert to Daniel Agger's flicked header.
The Reds' need to recover their form despite an injury list that includes Steven Gerrard and Fernando Torres and with disunity in the squad after Ryan Babel announced be planned to leave if he did not play more games.
Despite Benitez swapping the ineffective David Ngog for Netherlands winger Ryan Babel after 63 minutes, the Dutchman fared little better with the chances he was offered, squandering an excellent opportunity in front of goal four minutes after getting on the pitch.
Aquilani placed a shot just wide after an overhead ball from Ryan Babel was only half-cleared but Hull did create a chance of their own when Jan Vennegoor of Hesselink bulldozed through the Liverpool defence before heading straight at Pepe Reina.
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