Today’s draw in Nyon, Switzerland paired the losers of the sides who clash in Istanbul this week and then Scotland seven days later with the winners of the games between the Serbian and Austrian teams.
LASK will probably be the favourites to progress.
They were fourth in their league last season and were runners-up in their domestic cup competition.
Two years ago they got to the round of 16, when they were defeated by Manchester United. Last season they made it as far as the group stage, when their best result was a 3-3 draw with Spurs.
The matches are scheduled for August 19 and 26. The away game for Saints would be first.
The winners will progress to the group stage of Uefa’s new third tier competition, with big financial rewards to go with it.
Saints started their league campaign with a 0-0 draw against Ross County and will fly out to Turkey on Wednesday.