But the Dubliners scored a second on 102 minutes, Chris Fagan’s strike enough for them to seal a 3-2 aggregate win.
John Russell had put the hosts ahead just after the half hour, with Ivica Dzidic levelling on 65 minutes.
St Pat’s face a daunting clash with Bundesliga side Hannover in the third qualifying round.
Compatriots Sligo Rovers went out after a 1-1 draw at home to Slovakia’s Spartak Trnava, who won the first leg 3-1.
Northern Irish side Portadown had it all to do after a 6-0 first-leg loss to Slaven Belupo of Croatia, but they made a fist of it in losing the home leg 4-2.
Scotland’s St Johnstone took the lead against Turkish side Eskisehirspor, but they were held 1-1, leading to a 3-1 aggregate defeat.