St Patrick's appear to be heading out of Europe following a forgettable night on home soil, but Inter Milan, Athletic Bilbao and FC Twente picked up wins.
Anzhi and Heerenveen were also victorious in Europa League qualifying, but Marseille were held by St Johnstone's conquerors Eskisehirspor.
St Patrick's are staring elimination in the face after suffering a 3-0 defeat to Hannover in the first leg of their third qualifying round tie.
The Dublin side, who beat Bosnia's Siroki Brijeg in the last round, were up against it almost immediately after Denmark international Leon Andreasen put Hannover in front in the sixth minute.
Christian Pander doubled the visitors' lead in the 68th minute and reported Everton and Sunderland target Didier Ya Konan sealed an impressive win for the Bundesliga outfit 10 minutes from time.
Inter Milan cruised to a 3-0 victory over Hajduk Split, with Wesley Sneijder, Yuto Nagatomo and Philippe Countinho getting the goals for the Serie A giants.
Last season's runners-up Athletic Bilbao made light work of Croatian outfit Belup, with Ismael Lopez bagging a brace as they secured a 3-1 success.
Steve McClaren looks set to guide FC Twente through this stage of the competition after seeing off Mlade Boleslav 2-0.
Leroy Fer and Nacer Chadli were on target for the Dutch outfit against Czech opposition.
Another Eredivisie side to enjoy a productive night were Heerenveen, with Filip Djuricic grabbing a deadly double in a 4-0 stroll past Rapid Bucuresti.
Oleg Shatov and Fyodor Smolov were on target as Anzhi Makhachkala saw off Vitesse Arnhem 2-0, while Marseille were held 1-1 by Eskisehirspor - with Andre-Pierre Gignan on target for the Ligue 1 side.