Hajduk Split's late own-goal secured a 1-0 victory for Stoke in Croatia.
The Potters, who went into the third round second leg clash with a 1-0 lead from last week's first leg at the Britannia Stadium, put in an impressively controlled display over the 90 minutes at Stadion Poljud.
The performance was capped in stoppage time as Hajduk defender Ljubo Milicevic put Dean Whitehead's cross into his own net under pressure from Ryan Shotton.
Fulham moved through with a 2-0 victory over RNK Split at Craven Cottage.
After enduring a tough first leg in Croatia last week, the Londoners won a temperamental clash on home soil.
Andrew Johnson's header gave Martin Jol's side a first half lead, before skipper Danny Murphy wrapped the tie up from the spot.
Palermo were dumped out on away goals in the Europa League third qualifying round by Thun on Thursday night.
The first leg in Italy ended in a 2-2 draw and, though Pablo Gonzalez gave the Serie A side the lead, Dario Lezcano's goal saw the Swiss underdogs through.
Bundesliga side Mainz were another high-profile casualty as they lost on penalties to Romania's Gaz Metan.
The two sides had drawn their first leg 1-1 and the same scoreline was repeated on Thursday night, Marcel Risse giving Mainz the lead before Taher Bawab equalised.
Substitute Sami Allagui and Elkin Soto both missed from the spot for Mainz to allow captain Cristian Todea to settle the match in favour of Gaz Metan, for whom Alin Litu had failed to score.
Paulo Sergio steered Hearts into the final qualifying round with a 4-1 win on the night and 5-2 success on aggregate over Paks .
The new Hearts boss saw Ryan Stevenson score twice in 12 first-half minutes before Andrew Driver marked his return from injury with a third five minutes into the second half.
Rudi Skacel celebrated his new contract with a fourth before Daniel Bode scored an 89th-minute consolation for the Hungarians.
Irish duo Sligo Rovers and St Patrick's Athletic tumbled out of the competition.
Sligo appeared to have given themselves a chance of reaching the play-off round after holding Vorskla Poltava to a 0-0 draw away from home in the first leg.
However, there were only 16 minutes on the clock when Serhiy Zakarliuka put Vorskla ahead and just 60 seconds later the visitors made it 2-0 through Pavlo Rebenok, with no way back for Sligo after that.
The task for St Patrick's was more difficult after a 2-0 defeat away to Karpaty Lviv and a 3-1 home loss ensured a 5-1 aggregate scoreline.
Evan McMillan had drawn St Patrick's level on the night after Sergei Zenjov's opener, but Karpaty scored again through Ihor Khudobyak and Igor Oschypko.
Bursaspor survived a scare to qualify against Belarussian minnows Gomel .
The Turkish side led 2-1 from the first leg but Ilya Aleksievich put Gomel ahead on the night and in a position to qualify.
It stayed that way until the 78th minute, when Alfred N'Diaye restored Bursa's lead in the tie, and substitute Federico Insua and Turgay Bahadir then made it 5-2 on aggregate.
Atletico Madrid, who won the competition in 2009-10, will take their place in the play-off round after goals from Adrian Lopez and Jose Antonio Reyes secured a 2-0 win over Stromsgodset on the night and 4-1 on aggregate.
Denmark's Brondby suffered a surprise exit.
Ried went through on away goals despite a 4-2 defeat in Thursday night's second leg with Anel Hadzic's 88th-minute goal the decisive moment.
Victor Montano and Julian Feret scored as Rennes won 2-0 to complete a 7-2 rout of Metalurgi Rustavi , and Dinamo Tbilisi were equally comfortable as they also won 2-0 to beat KR Reykjavik 6-1 in the tie.
Helsingborg overturned a 1-0 first-leg deficit to beat Israel's Bnei Yehuda 3-1, Mattias Lindstrom scoring twice and Ardian Gashi with the other goal on the night.
Salzburg won 3-0 at Senica , Leonardo setting them on their way before Alan's brace wrapped up a 4-0 aggregate success.
Compatriots Austria Vienna held their nerve to see off Olimpija Ljubljana 3-2 after a 1-1 first-leg draw.
Nacer Barazite had put them in control at 3-1 on aggregate but the Slovenians levelled the tie through Dare Vrsic and Boban Jovic, only for Barazite to complete his hat-trick with a penalty and ensure his side's continued participation in the competition.
AZ Alkmaar of the Eredivisie comfortably wrapped up a 3-1 aggregate win over Jablonec after drawing their second leg 1-1.
Dinamo Bucharest beat Varazdin 2-1 to secure a 4-3 aggregate success and Club Brugge beat Qarabag 4-2 on aggregate despite a 1-0 defeat on Thursday night.
Andrija Kaludjerovic and Cristian Borja scored twice as Red Star Belgrade beat Ventspils 7-0 on the night and 9-1 in total, while Sparta Prague beat Sarajevo 2-0 to wrap up a 7-0 result.
Tel Aviv rivals Maccabi and Hapoel both qualified along with AEK Larnaca, Aalesund, Spartak Trnava, Nacional, Slask Wroclaw, Young Boys, Omonia Nicosia, Olympiacos Volou, Rabotnicki Kometal, PAOK Salonika, Legia Warsaw, Alania Vladikavkaz and Vitoria Guimaraes.