On a night featuring 35 ties littered with some of the biggest names in European club football and many obscure ones trying to make a name for themselves, Liverpool gained a narrow advantage to take to Turkey next week when the winners on aggregate will move into 48-team group phase.
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Italian giants Juventus, who like Liverpool find themselves chasing silverware in the continent's secondary competition after failing to qualify for the Champions League, sneaked a 2-1 victory in Austria against Sturm Graz.
Porto, CSKA Moscow, Bayer Leverkusen, Paris Saint-Germain, Borussia Dortmund and Celtic all scored comfortable first leg victories while big-spending Manchester City got their European season up and running with a 1-0 away victory against Romanian side FC Timisoara - Mario Balotelli scoring on his debut after signing last week from Inter Milan.
One of the night's surprises saw Dutch club PSV Eindhoven, winners of the trophy in its previous incarnation as the UEFA Cup in 1978, lost 1-0 in Russia against Sibir Novosibirsk.
Cole, who was sent off during his Liverpool debut against Arsenal in the Premier League on Sunday after moving on a free transfer from Chelsea, had his second-half penalty saved by Trabzonspor keeper Onur Kivrak.
That would have given Liverpool, England's most successful team in European club football, some breathing space but they had to settle for the solitary goal scored by Ryan Babel.
GOOD VALUE
"I think particularly in the second half we were good value for our victory and I think if you miss a penalty you only have yourselves to blame," Liverpool manager Roy Hodgson, who led Fulham to the Europa League final last season, told ITV.
Liverpool's other new signing, Christian Poulsen, had a goal ruled out in an eventful second half.
Former Liverpool striker Milan Baros scored twice to earn Turkish club Galatasaray a 2-2 draw with Ukraine's Karpaty Lviv.
Twice European champions Juventus had a stoppage time goal by Amauri to thank for their victory over Sturm Graz after Leonardo Bonucci's first-half header had been cancelled out late on by Gordon Schindenfeld.
Italy's other representaives Napoli and Palermo both won their first legs against Elfsborg Boras and Maribor respectively.
Porto enjoyed an impressive 3-0 away win at Belgium side Racing Genk while CSKA Moscow made the second leg of their tie a formality after thrashing Anorthosis Famagusta of Cyprus 4-0.
