The second leg in Denmark next week would have been an irrelevance had Fortuna not missed a penalty.
However, City's second half effort, when Jane Ross score her 150th goal for the club, gave the Scots a glimmer of hope.
"The first half cost us," said City coach Eddie Wolecki Black.
"If Claire Johnstone had not saved a penalty in first half the tie was finished.
"The girls are disappointed with the defensive aspect.
"But we are capable of scoring goals over there. The big question is whether we lose any more silly ones."