With a 6-0 lead from the first leg, John Barnes was able to give six players their first start of the season, and one of them, Norway's Harald Brattbakk, broke the deadlock after just nine minutes when he collected a Regi Blinker pass and slotted the ball past the onrushing goalkeeper Stephen Morris.
The 19-year-old substitute Jamie Smith doubled the Glasgow side's lead with half an hour remaining when he cut inside full-back John Wills and curled a shot into the top corner. Johann Mjallby made it 3-0 when he got on the end of a cross from Smith, and Tommy Johnson concluded the rout in the final seconds.
Two goals in the last three minutes by the substitute Miguel Simao helped St Johnstone overcome Finland's VPS Vaasa 2-0 on the night and 3-1 on aggregate in their qualifying tie at McDiarmid Park.
After coming on to replace Nathan Lowndes, the Portuguese striker latched on to Ged McMahon's pass to drill the ball home past the Vaasa goalkeeper Henri Sillanpaa, and then he wrapped things up in injury-time after taking the ball round Sillanpaa for his second.
Kilmarnock produced a last-ditch recovery act to survive being upstaged by 10-man Reykjavik at Rugby Park.
The Ayrshire club were on the verge of removal from the competition against the Icelandic part-timers, who held their 1-0 first-leg advantage until the 90th minute. That was the moment Einar Pro Danielsson's handball was punished by Paul Wright when he neatly converted the spot-kick.
Two minutes into extra-time a stunning individual effort by the substitute David Bagan put Kilmarnock into the hat for today's first-round draw.