Babel, playing as a lone striker for a much-changed Reds side, struck in injury time of an otherwise dismal first half at Anfield.
Liverpool should have won more convincingly but big-name new signing Cole, sent off against Arsenal on Sunday and found guilty of speeding on Wednesday, missed a second-half penalty.
Debutant Christian Poulsen also had a goal disallowed as Liverpool carried a much greater threat after the break but the contest remains far from over with another 90 minutes to play.
Manager Roy Hodgson gave Steven Gerrard the night off and Liverpool’s dynamic captain was conspicuous by his absence throughout a drab opening 45 minutes.
Little of note occurred until Cole slipped in Babel on goal with a fine ball in stoppage time at the end of the first half and the Dutchman calmly clipped over the advancing Onur Kivrak.
Prior to that a Liverpool side retaining only four of the XI which opened the Barclays Premier League season against Arsenal lacked drive and direction.
Liverpool showed patience on a surface made greasy by persistent rain and had a chance after five minutes as Sotirios Kyrgiakos tested Kivrak with a bouncing header.
Yet it was a rare moment of incision in the first period as both sides were restricted to long-range, and invariably off-target, free-kicks.
The more fluid attacks came from Trabzonspor and they gave Liverpool a scare when Reds right-back Martin Kelly was forced to head past his own post from a free-kick.
Selcuk Inan then flashed a header across goal from the resulting corner.
Yet neither goalkeeper had a save to make until Liverpool suddenly woke from their slumber with 45 minutes on the clock.
Cole found space and picked out the onrushing Babel with precision and the Holland international struck clinically.
Unfortunately for Babel, linked with a move to Wolfsburg this week, it proved to be his final touch of the game as he was replaced by Torres at the interval.
The Spaniard, feeling his way back into action after a groin injury, needed little time to show his class as he tested Kivrak with a long-range shot after a driving run.
Liverpool looked a different team from the one that had played out most of the first half and an unmarked Lucas headed a glorious chance wide from a Milan Jovanovic corner.
Poulsen then won possession and combined well with Jovanovic before lashing another effort over.
Liverpool should have had a penalty when Rodriguez was felled in the area after 50 minutes but the referee pointed to the spot barely a minute later after Serkan Balci brought down Lucas.
Cole, sent off against Arsenal and anxious to get his Anfield career up and running, stepped up to take the spot-kick but shot tamely at Kivrak.
Liverpool were almost made to pay just before the hour as Umut Bulut showed great control on the edge of the area to turn and shoot but Reina deflected the ball over.
Torres, looking a different player from the one who laboured through the World Cup, rose to meet a Jovanovic cross but Kivrak grasped his firm header.
Liverpool thought they had claimed a second after 72 minutes when Kivrak spilled a Kyrgiakos header and Cole nodded on for Poulsen to stab in.
The Dane, however, had inched offside and the goal was ruled out.
Aurelio then found good space but dragged a shot wide from 12 yards.
For all Liverpool’s dominance, the tie was still far from safe ahead of next week’s return trip to Trabzon.
The visitors played out the closing minutes safely, perhaps hoping for another chance on home soil.