The New Saints saw their hopes of Champions League progress dented after a 2-0 away defeat against Croatian side Rijeka in the first leg of their second qualifying round tie.
The Welsh Premier League side were soon trailing in Croatia when midfielder Josip Misic netted a fine 20-yard free kick, clipping the ball over the wall and into the top corner in the fifth minute.
Rijeka, who were given a bye straight into the second qualifying round, doubled their advantage through substitute Florentin Matei after 69 minutes.
Quigley lashed a late angled shot into the side-netting as TNS were left with it all to do in the second leg at Park Hall.
In Tuesday's other games, Austrian champions FC Salzburg cruised to a 3-0 win away to Malta's Hibernians.
Valon Berisha opened the scoring with a penalty in the 32nd minute, before Hee-Chan Hwang doubled the lead three minutes later. Munas Dabbur added a third in the second half, though Berisha was sent off late on.
Salzburg debuted a new shirt that is remarkably similar to that of sister club RB Leipzig, despite UEFA clearing both clubs to enter the competition after ruling that they were sufficiently separate entities.
Serbia's Partizan Belgrade defeated Montenegro's Buducnost Podgorica 2-0 with second-half goals from Uros Djurdjevic and Leonardo.
Qarabag, the Azerbaijani club that reached the Europa League group stage last year, cruised to a 5-0 win at home over Georgia's Samtredia, thanks to a brace from Dino Ndlovu.
The second round continues on Wednesday with 12 more games, before Linfield host Celtic on Friday.
Information from Press Association was used in this report.