Barcelona, helped by a hat-trick from Brazilian forward Ronaldinho, two apiece from Luis Enrique and Javier Saviola and another goal from Thiago Motta, crushed Slovakia's Puchov 8-0 at the Nou Camp to go through 9-1 on aggregate.
Liverpool enjoyed a comfortable passage at Anfield, thumping Slovenia's Olimpija 3-0 for a 4-1 win over the two legs.
But the night did not pass without incident off the pitch as English and Polish fans were arrested in the Netherlands.
Dutch police said 94 fans, 87 of whom were said by police to be Newcastle United supporters, were arrested in Breda after riots broke out in the city centre before NAC's match.
Dutch police also arrested 41 Polish fans of Wisla Krakow earlier on Wednesday after they found two axes, a garotte, three switchblades, kitchenknives, and other weapons on their bus travelling to the game against NEC Nijmegen.
Barcelona had not previously won at the Nou Camp this season and their slow start had prompted coach Frank Rijkaard to call for patience from the club's discontented fans.
But Ronaldinho, who had been doubtful for the game after picking up a leg strain, silenced the doubters in a paltry 15,000 crowd with an early goal after seven minutes.
He added a second 13 minutes later, and a third goal four minutes before the break from Thiago Motta put Barca's passage beyond doubt after their surprising 1-1 first-leg draw.
Ronaldinho completed his hat-trick on 57 minutes before two each for Luis Enrique and Saviola completed the rout.
Liverpool also made a quick start against Olimpija of Slovenia. French teenager Anthony Le Tallec, who joined from Le Havre in the close season and was replacing injured striker Michael Owen, scored his first goal for the club on 30 minutes.
Emile Heskey added a second shortly after and Harry Kewell volleyed home the third goal two minutes into the second half.
Borussia Dortmund, the 1997 European champions, also eased through against Austria Vienna, midfielder Lars Ricken giving the German side a 1-0 home win for a 3-1 aggregate victory.
But it was a miserable night for two other Bundesliga sides, with Hertha Berlin and Hamburg SV dumped out of the competition.
Hertha, struggling in the league after failing to win in their first eight games, went out to Poland's Groclin Grodzisk, increasing the pressure on coach Huub Stevens.
An 84th minute goal from Grzegorz Rasiak gave Groclin a 1-0 home win after a goalless first leg.
Hamburg, also in the lower half of the German first division, were hammered 3-0 in the Ukraine by Dnipro Dnipropetrovsk to go out 4-2 on aggregate.
The 1983 European champions fell behind to an early goal from Dmytro Mikhailenko and strikes after the break from Olexander Rykun and Oleg Venglinsky sealed their fate.
Parma and Perugia progressed at the expense of Metalurg Donetsk and Dundee respectively, but fellow Serie A side Udinese lost 2-1 at home to Austria's Salzburg and went out on the away goals rule.
Turkey's Genclerbirligi gained a modicum of revenge for the national side's failure to qualify automatically for next year's Euro 2004 finals after a goalless draw with England on Saturday by putting out premier league side Blackburn Rovers.
Genclerbirligi, leading 3-1 from the first leg in Ankara, fell behind to a Matt Jansen goal on 65 minutes but Mustafa Ozkan earned a 1-1 draw with an equaliser a minute later.
Harald Brattbakk hit a hat-trick for Rosenborg Trondheim who romped to a 6-0 win over Latvia's Ventspils to complete a 10-1 aggregate victory.
The Norwegian champions are playing in the UEFA Cup after failing to qualify for the group stages of the Champions League for the first time in eight seasons.
Swiss league leaders Basel scraped into the second round thanks to a 'silver goal' from Marco Streller against Turkey's Malatyaspor.
Basel, who have chalked up a record 13 straight wins in their domestic league, won the first leg 2-0 away from home but two goals from Celalettin Kocak on Wednesday levelled the tie and sent the game into extra-time.
Streller's goal five minutes into the first extra period put the Swiss side back in front on aggregate and they held on until the half was over, when the match finished under UEFA's 'silver goal' rule.