Quigley scored TNS's equalising goal against the Gibraltarian club after Alex Quillo had given the visitors the lead eight minutes in.
On the stroke of half-time, the 24-year-old forward latched onto a through-ball before deftly flicking it over the opposition defender with one touch on the turn, and then rifling a shot past the goalkeeper from a difficult angle into the roof of the net.
Europa snatched victory through Quique Gomez in the 78th minute, meaning they hold a strong advantage heading into the second leg next week.
TNS have reached the second round of qualifying in each of the last five seasons, so will be hoping Quigley can produce some more magic on the away trip.
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