United, roared on by nearly 1,000 fans who had made the long journey, started the game the better and should have taken the lead as early as the third minute but the header from Bert Esselink was high over the bar.
Strassen were playing in only their fourth European match and had failed to find the net in their previous three matches.
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 of lapses at the back.
United had more chances. Craig Sibbald won the ball in midfield and set up Kristijan Trapanovski down the right-hand side and his cross fell perfectly for Zac Sapsford, who fired over from just eight yards with the goal gaping.
And the United fans were on their feet again when Ivan Dolcek hooked a volley goalwards, but home keeper Koray Ozcan dived high to his right to push the ball away.
There was a blow for United before the break when Trapanovski had to be replaced with what looked like a hamstring injury and Max Watters took his place.
The hosts continued to create chances and Matheus de Souza pulled his right-foot shot wide from 16 yards with the United defence struggling to get back to cover.
However, it was the Scots side 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 nodded 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.