United, roared on by nearly 1,000 fans, started well and should have taken the lead as early as the third minute but a header from Bert Esselink was high over the bar.
Strassen were playing in only their fourth European match and have yet to score.
But slowly the hosts began to find space and almost caught out the United defence on a couple of occasions, with Daryl Myre failing to take advantage.
United had more chances. Craig Sibbald won the ball in midfield and set up Kristijan Trapanovski down the right and his cross fell perfectly for Zac Sapsford, who fired over from just eight yards with the goal gaping.
The United fans were on their feet again when Ivan Dolcek hooked a volley goalwards, but home keeper Koray Ozcan dived high to push the ball away.
The hosts continued to create chances and Matheus de Souza pulled a shot wide from 16 yards.
However, it was the visitors who would get the solitary goal in the 63rd minute to put the tie to bed. Sapsford's floated free-kick to the back post was met by Moldovan defender Iovu, who steered the ball back across goal and into the bottom corner.
The tension suddenly eased for United, with the large travelling support finally able to enjoy their trip.
Michael Stewart
