Swiss club Grasshoppers beat MyPa of Finland and Bulgarian outfit Litex Lovech defeated Belgian side Genk.
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Welsh side TNS have a tough match against MyPa of Finland, with the winners playing FC Copenhagen.
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"This has been a problem and it will continue in the early rounds of the season, " said MyPa coach Ilkka Makela.
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Blackburn needed the cushion of a second goal to settle any nerves, and twice MyPa almost coughed one up free of charge.
Finnish champions MyPa will be missing star strikers Saku Puhakainen and Niki Helenius in their first-round games set for 12 July and 19 July.
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The Saints, who lost to Finland's MyPa-47 at the first hurdle last year, will face Austrians FC Salzburg if they advance from the two-leg first round.
MyPa, who dumped Scottish Premier League side Dundee United out of last season's Uefa Cup second round and have also beaten Motherwell, were the second seeds in Friday's draw.
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The 3, 000 crowd even took time out from fending off mosquitoes to get behind their team as MyPa grew in confidence, and Christopher Samba was forced into a last-ditch tackle to deny Eero Peltonen a shot at goal.
Blackburn had to endure something resembling a spell of pressure straight after the restart, with MyPa forcing a series of corners, although at no time was Brad Friedel in the Rovers' goal required to make any sort of save.
Blackburn were so far ahead of the Finnish outfit in terms of technique and tactical nous that the early exchanges were little more than a practice match in MyPa's bucolic little ground which resembled most Premier League clubs' training facilities.
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