St Patrick's booked their spot thanks to Christy Fagan's extra-time winner against Siroki Brijeg.
Fagan struck in the 105th minute to seal a 2-1 win on the night and a 3-2 victory on aggregate for Liam Buckley's side, who now face German club Hannover.
St Patrick's had earlier seen the 39th-minute lead given to them by John Russell cancelled out by Ivica Dzidic's 65th-minute equaliser, sending the tie to extra time after the first leg had also finished 1-1.
It was not such a good night for Sligo and Portadown, though, with both bidding farewell to the competition.
St Patrick's' domestic rivals Sligo were knocked out 4-2 on aggregate by Slovakian outfit Spartak Trnava after only managing a 1-1 draw at The Showgrounds in Thursday night's second leg.
Peter Cvirik gave Spartak a 70th-minute with Jason McGuinness' equaliser in stoppage time for Sligo.
Northern Irish side Portadown were sent tumbling after a comprehensive 10-2 aggregate defeat at the hands of Slaven Koprivnica.
The Croatian side had done all the hard work in the first leg as they took a 6-0 lead for the trip to Shamrock Park.
Goals from Igor Bubnjic (10), Petar Brlek (14) and Martin Saban (62 and 64) earned the visitors a 4-2 victory on Thursday night.
Portadown's replies came from Richard Lecky (45) and Neil McCafferty (67).
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