Derry City, backed by a rapturous home crowd, dominated the opening exchanges of the contest.
The Candystripes really should have levelled the contest on aggregate early on when Michael Duffy and Ellis Chapman combined to set up Liam Boyce.
However, the striker watched his effort from 10 yards saved by the outstretched Aleksa Todorovic.
Derry were made to rue their missed opportunities as ex-Bournemouth striker Adu-Adjei latched onto a poor back-pass from Cameron Dummigan and raced clear to finish superbly beyond Brian Maher to double the visitors' aggregate advantage after just 14 minutes.
The goal knocked the stuffing out of Lynch's side as the Croatians started to get into their groove and control both the ball and the tempo of the contest.
The Candystripes needed a spark in the second half but it never arrived and the visitors could have doubled their lead when Adu-Adjei shot wide before heading off target from point-blank-range.
Duffy had a chance from a free-kick, but Derry's European exploits ended with a relative whimper.