Finland's MyPa doubled their first-leg advantage after six minutes when Saku Puhakainen latched on to a long ball and lobbed goalkeeper Gerard Doherty.
TNS only tested the MyPa keeper once through a Greg Stones half-volley.
Steven Beck wasted a glorious chance to grab a draw when he put a penalty wide of the left-hand post in injury-time.
TNS manager Ken McKenna:
"It looked a pre-season performance from the lads, and that's what it was because we haven't got any competitive games under our belts. "We weren't sharp, they were a bit sharper and we were caught cold by the early goal and then it's catch-up.
"Over the two legs we should've got something out of it, but we didn't have it in our legs to open them up."
TNS: Doherty, Baker, King, Jackson, Holmes, Ruscoe, Hogan, Wilde, Beck, Ward, Stones.
Subs: Williams, Leah, Toner, Carter, Davies, Harris, Williams.
MyPa: Korhonen, Timoska, Taipale, Lindstrom, Kuparinen, Puhakainen, Manso, Leguizamon, Mirands, Kansikas, Adriano.
Subs: Tuomaia, Hemesniemi, Huttunen, Kangaskolkka, Juura, Oksnaen, Peltonen
Referee: Peter Vervecken (Bel)
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