Forward Stephen O'Flynn proved the hero for the Irish club, scoring in extra-time to send the Dubliners through on the away goals rule, having won 1-0 at home in the first leg.
Matt Garner - one of a number of Englishmen in the Icelandic team's line-up - struck late on to break the deadlock on the night and draw the hosts level on aggregate, sending the tie into an additional period.
Eyþór Helgi Birgisson then grabbed the advantage for IBV when he swooped in the eighth minute of extra-time but St. Pat's hit back immediately through O'Flynn just a minute later to knock IBV side out of the competition for the second year on the run.
St. Pat's now face Siroki Brijeg from Bosnia-Herzegovina.
Otherwise, it was a poor night for British and Irish sides as six other clubs crashed out of the competition.
Welsh trio Llanelli, Bangor City and CEFN Druids all went out after decent first-leg results, while Northern Irish duo Cliftonville and Crusaders both exited.
Ireland’s Bohemians were hammered by a second division side from Iceland.
There was heartbreak for Llanelli of Wales as they lost narrowly to Finnish side KuPS, a 1-1 home draw resulting in a 3-2 aggregate defeat for the part-timers.
MyPa Myllykoski sealed a Finnish double over Welsh opposition as they beat CEFN Druids 5-0 after being held to a creditable goalless draw in the first leg.
The third Welsh side in action, Bangor City, suffered a narrow 2-1 defeat to Moldova’s Zimbru Chisinau, again after a goalless draw in Wales.
Cliftonville of Northern Ireland had won the home leg 1-0 over Kalmar, but the good work was undone by a 4-0 away loss to the Swedes.
Fellow IFA Premiership side Crusaders also lost, although the 1-0 defeat at Norwegian big-guns Rosenborg was a better result than their 3-0 home defeat.
To teams south of the border, Bohemians were thrashed 5-1 by second division KA Akureyri in Iceland, having held them to a goalless draw in the first leg.