Roma left out their big names for the third round, first leg match against Gaziantepspor and lost 1-0 in southern Turkey while financially-troubled Parma slumped by a similar scoreline at home to Genclerbirligi.
Three-times UEFA winners Inter Milan, despatched to this competition after finishing third in their Champions League group, restored a modicum of pride but having twice led in France against Sochaux, Alberto Zaccheroni's side were pegged back for a 2-2 draw.
Former European champions Barcelona will be confident of progressing after Ronaldinho's goal earned a 1-0 win in Denmark over Brondby, while Liverpool, winners in 2001, eased the pressure on under-fire coach Gerard Houllier by defeating Levski Sofia 2-0 at Anfield.
A 19th-minute goal from on-loan midfielder Yusuf Simsek handed Gaziantepspor, sixth in their domestic league and easy winners over French side RC Lens in the previous round, a narrow advantage to take to the Olympic Stadium.
Roma coach Fabio Capello left the likes of Francesco Totti, Antonio Cassano and Vincenzo Montello at home and the Turkish side took full advantage. Roma's cause might have been made harder in the second leg but Ghanaian Samuel Johnson struck the crossbar late on.
Parma, the 1995 UEFA Cup winners who face an uncertain future after their owners Parmalat were forced into insolvency in December, fell behind to a goal on the hour from Genclerbirligi midfielder Josip Skoko and wasted chances to equalise.
"I am satisfied with the result and we could have won by a larger margin because we were clearly the better side," said Genclerbirligi coach Kazim Ersun Yanal.
Inter Milan striker Christian Vieri shrugged off Tuesday's fire bomb attack on a restaurant he co-owns by netting after just eight minutes against high-flying Ligue 1 side Sochaux.
Inter led again in the second half but two goals from striker Pierre-Alain Frau gave Sochaux hope for the return leg.
There was no fairytale victory for Brondby coach Michael Laudrup as his former side Barcelona produced a professional performance in the Danish capital.
A goal four minutes into stoppage time by David Navarro gave Valencia, among the competition favourites, a 3-2 win over Turkish league leaders Besiktas, who had threatened to cause a shock at Mestalla.
Romanian Daniel Pancu twice fired Besiktas ahead but goals from Mohamed Sissoko and Fabian Canobbio put the Spanish team level before Navarro's late winner.
Liverpool's poor domestic form has led to calls for Houllier's resignation but the Frenchman was smiling after second-half goals from Steven Gerrard and Harry Kewell broke down resilient Levksi.
"I was moved by the chants," Houllier said after the home fans rose to acclaim him in a show of support late in the game.
"A lot of the criticism has been unfair. We need to be patient."
Celtic began their quest to reach back-to-back UEFA Cup finals by cruising to a 3-0 home win over Czech minnows Teplice.
Two goals from Henrik Larsson and another from Chris Sutton gave the unbeaten Scottish premier league leaders, who lost last year's final 3-2 to Porto, a comfortable advantage.
Former European champions Marseille edged out Ukraine's Dnipro Dniepropetrovsk 1-0, Real Mallorca romped to a 3-0 win at Spartak Moscow while fellow Spaniards Villarreal, rocked by the resignation of coach Benito Floro on Monday, drew 2-2 in Istanbul against 2000 winners Galatasaray.