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.
Their win came 10 days after the group's LP Babel took the album of the year trophy at the Grammy Awards in Los Angeles.
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.
Mumford and Sons' second effort Babel rested at number two after climbing two places since the folk rockers won best British group at the awards ceremony.
Liverpool quelled the growing Napoli tide as the game opened up and Babel had a great opportunity after again being played in by Jovanovic but had his side-footed attempt saved by the foot of keeper Morgan De Sanctis.
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.
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.
The touch paper had well and truly been lit and, after Hyypia had penalty appeals waved away following a Drogba challenge, Babel reduced the arrears with a speculative effort from 35 yards that Cech should have done better with.
But Babel wasted a glorious opportunity to snatch the win for Liverpool after 88 minutes, when he somehow headed Kuyt's cross wide with the goal at his mercy.
"He (Babel) will have a chance to play, " said Benitez after his side's 1-1 FA Cup draw with Reading.
He found Babel unmarked only eight yards out, and he slammed home an emphatic finish after Green blocked his first attempt.
After the break, Liverpool pressed for another goal and they might have had one had Babel released Keane rather than opted to shoot.
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.
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