Hearts of Midlothain were shocked 2-1 at home by Malta's Birkirkara after a goalless first leg. Christian Bubalovic and Edward Herrera's goals rendered Conor Sammon's reply meaningless.
Kevin O'Connor's first-half goal gave Cork a 1-0 win at home to Swedish side Hacken and secured a 2-1 aggregate win which put them into the third qualifying round of the Europa League.
Aberdeen beat Ventspils 1-0 with Adam Rooney on target, completing a 4-0 aggregate win, but there were defeats for Edinburgh rivals Hibernian.
Hibs beat Brondby 1-0 in their second leg, David Gray's goal levelling the tie at 1-1, but lost 5-3 on penalties as John McGinn's opening effort was saved.
Cliftonville's European involvement ended in Cyprus where AEK Larnaca won 2-0 thanks to goals from Vladimir Boljevic and Joan Tomas to secure a 5-2 aggregate win.
The Belfast side had Caoimhin Bonner sent off after just 11 minutes.
The Welsh Premier League's Connah's Quay exited 3-1 on aggregate after Vojvodina extended their 1-0 first-leg lead with two goals from Dejan Meleg, the second a penalty, and Michael Wilde's response midway through the second half could not spark a fightback.
St Patrick's Athletic's creditable 1-1 draw in Belarus last time out counted for nothing as Dinamo Minsk progressed thanks to Gleb Rassadkin's solitary goal in the Dublin return leg.
For more results from Thursday's Europa League action, click here.