The Lilywhites went 1-0 down against FH Hafnafjordor on the night but two goals from David Mc Millan gave them a 2-1 lead before a Finbogasson equaliser.
Mc Millan's goals coming shortly after Ronan Finn's penalty was saved.
The match ended 2-all on the night and 3-all on aggregate meaning Dundalk go through on away goals.
Dundalk go to Bate Borisov of Belarus in the first leg next Tuesday with the return leg at Tallaght stadium a week later as Oriel Park is not up to standard for European games at this stage.
Celtic are also through to the next stage having beaten Lincoln Red Imps of Gibraltar last night 3-0.
The Hoops will play Astana of Kazakhstan.
