In a stunning Europa League upset in the Stadio Olimpico, Peter Stepanovsky scored in the 82nd minute to give Slovan Bratislava a 1-1 draw on the night with Roma and a 2-1 aggregate win.
Simone Perrotta had cancelled out Slovan's slim first-leg advantage with an 11th minute effort, but the home side failed to build on that lead and Stepanovsky's strike embarrassed the Serie A giant.
There were no such problems for Roma's city rival Lazio, who wrapped up a 3-1 win over Rabotnicki in Macedonia for a 9-1 aggregate victory, with Tommaso Rocchi scoring twice.
Sevilla was knocked out after a 1-1 home draw with Bundesliga side Hannover. Mohammed Abdellaoue gave the German side the lead and an Emanuel Pogatetz own goal was not enough for the Spaniards, who went out 3-2 on aggregate.
Klaas-Jan Huntelaar scored four goals including two penalties as Schalke blasted back from a two-goal first-leg deficit to hammer HJK 6-1 and reach the Europa League group stage.
Atletico Madrid left wantaway striker Diego Forlan at home but it made little difference as the Primera Division eased to a 4-0 win at Vitoria for a 6-0 aggregate win, with Adrian scoring twice.
PSV Eindhoven shrugged off last week's goalless draw in Austria by thumping Ried 5-0 at home.
Last year's losing finalist Braga scraped through on away goals after a 2-2 draw with Young Boys of Switzerland. Former European champion Red Star Belgrade was thumped 4-0 in France by Rennes for a 6-1 aggregate loss.
All three Scottish sides crashed out while all the English sides involved secured a group-stage spot.
But the big story of the night involving British and Irish sides came in Belgrade where Stephen O'Donnell's extra-time penalty helped Shamrock Rovers make history as the first Irish side to reach the group stages of a European competition with a famous win over Partizan.
Substitute O'Donnell slid home a 113th minute spot-kick to give Rovers a 2-1 win on the night, enough to take them through 3-2 on aggregate after last week's draw in Dublin.
But while the Irish side celebrated, Rangers, Celtic and Hearts all crashed out, meaning there will be no Scottish involvement in the group stages of either the Champions League or Europa League.
Rangers, trailing Slovenian champions Maribor 2-1 from the first leg, spurned a host of chances at Ibrox before Dalibor Valas extended the visitors' aggregate lead.
Carlos Bocanegra restored Rangers' hopes with an equaliser on 75 minutes, but the hosts couldn't find the goal to force extra-time.
Celtic never recovered from falling a man and goal down within three minutes at FC Sion, eventually falling to a 3-1 defeat both on the night and on aggregate.
However, Celtic are looking for a red tape reprieve as Sion had five players in their squad - Feindouno, Mario Mutsch and Gabri, Jose Goncalves and Billy Ketkeophomphone - who are, in the eyes of FIFA, ineligible to play in the competition.
Sion's European squad, though, was approved by the Swiss FA who, along with the club, have reportedly been warned by UEFA general secretary Gianni Infantino about possible sanctions although the Court of Arbitration for Sport, who will rule on the issue, will not pass their verdict next week.
A second-string Tottenham side played out a tame goalless draw with Hearts at White Hart Lane to ease through to the group stage after last week's 5-0 victory at Tynecastle.
Championship side Birmingham defeated Nacional 3-0 at St Andrews to seal an aggregate victory by the same scoreline
Fulham is safely through despite losing 1-0 at Dnipro, the Cottagers' 3-0 win in last week's home leg securing their passage 3-1 on aggregate.
Stoke cruised into the group stages as Kenwyne Jones helped them wrap up a comfortable 5-1 aggregate victory over FC Thun with a brace in the play-off second leg at the Britannia Stadium.
Poland's Legia Warsaw staged a stunning comeback to beat Spartak Moscow 3-2 in the Russian capital for a 5-4 aggregate win. Legia trailed 2-0 on the night but capped their recovery with Janusz Gol's injury-time winner.
Two early goals from Joseph Akpala saw Club Brugge to a 2-0 win over Zestafoni and 5-3 aggregate success, while Belgian rival Anderlecht edged through 4-3 on aggregate after a 2-2 draw with Bursaspor.
French Ligue One side Sochaux was stunned 4-0 at home by Metalist after last week's goalless draw in Ukraine, but Paris St Germain wrapped up a 6-0 aggregate win over Differdange of Luxembourg with a 2-0 home stroll.
Dutch side AZ overturned last week's 2-1 reversal to Norway's Aalesund in style, winning 6-0 in Alkmaar. Martin Hinteregger's strike was not enough for Salzburg, who went out on away goals to Omonia Nicosia despite a 1-0 win on the night.
