Maxi Rodriguez hit the bar in the first-half, while Ngog missed three chances after replacing Torres.
Cole was red-carded for a lunge on Laurent Koscielny before David Ngog's driven shot put Liverpool ahead.
Yossi Benayoun slotted in a second to secure the win after keeper Marton Fulop had spilled Ngog's cross.
Birmingham hit back as Liam Ridgewell bundled in James McFadden's cross and survived three late David Ngog chances.
Liverpool finally found some rhythm and David Ngog had a chance to equalise but headed straight at Andreas Isaksson.
Torres was replaced on 65 minutes by David Ngog, who missed several chances as Liverpool pushed for a vital win.
David Ngog swivelled and had a shot blocked by Napoli's Salvatore Aronica as both teams had to settle for a draw.
They have already drafted in Philipp Degen, fellow full-back Andrea Dossena, goalkeeper Diego Cavalieri and teenagers David Ngog and Emmanuel Mendy.
Benitez introduced David Ngog and summer-signing Alberto Aquilani in the latter stages but Arsenal close out the latter stages in relative comfort.
Shelvey drifted a cross to the far post which Ngog could only steer harmlessly across the Napoli box rather that at goal.
Meireles slipped a ball across to Ngog, whose attempt to turn in a shot from close range was blocked by keeper De Sanctis.
Sunderland tried to muster a response but struggled retain possession before Fulop's error from Ngog's hooked cross allowed a grateful Benayoun to slot home.
They were eventually made to pay for the miss when David Ngog side-footed in his first league goal for the Reds after Steven Gerrard's headed pass.
And Liverpool missed the Spanish international's cutting edge as they dominated, Ngog lashing a half-chance high and wide after being put through by Javier Mascherano early on.
Ngog came storming into the six-yard box but, faced with an open goal, contrived to shin the ball against the crossbar, with Kuyt's follow-up diverted to safety.
Gerrard produced a wonderful piece of magic on the right to find Dirk Kuyt who poked in for 3-0 before substitute David Ngog nodded in in injury time.
Benitez reacted by bringing on Ngog for Torres, who walked off the pitch seemingly perplexed at his manager's decision, while Gerrard also appeared to be a little confused.
"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.
And with almost the last touch of the game, substitute Ngog made the most of Sorensen's inability to deal with a deflected cross to head in under the bar.
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.
Poulsen tried to make amends for his error be slipping a ball through to Ngog but the young striker could only side-foot wide under pressure from ex-Reds left-back Dossena.
It seemed to be developing into a familiar story for the Reds but Frenchman Ngog made the vital breakthrough for the home side after Gerrard steered Riera's cross into his path.
Ngog had the arduous task of leading the Liverpool line almost single-handedly and, while he worked tirelessly, he failed to make the most of a rare chance which came his way.
But, there were no signs that Liverpool were feeling the effects of their exertions in the week as they started brightly with chances falling to David Ngog, Maxi Rodriguez and Aquilani.
However the substitution introduced new venom to the Liverpool attack and Ngog, the striker who won a controversial penalty against Birmingham in November, went close twice with a header and shot.
The midfielder cajoled his stuttering side into a greater urgency and, after David Ngog and Raul Meireles missed two gilt-edged chances, it was left to the Reds captain to grab an equaliser.
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.
French striker David Ngog scored Liverpool's early winner, but Rafael Benitez's side struggled to produce a convincing display, the damage having been done in two games against Lyon that cost them four points.
Napoli defender Salvatore Aronica was fortunate to stay on the pitch for a tackle on Ngog but Liverpool will feel justice was done when the Italian was punished for bringing down Reds right-back Johnson.
Arsenal were shaken and the 10 men of Liverpool seized the initiative, with Ngog wasting a promising chance to add to the lead when he headed a Steven Gerrard free-kick over from six yards.
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