Aaron Callaghan’s side had much of the ball and created the better chances throughout, also having a goal disallowed. But they rode their luck a little at time as Thor had a decent claim for a penalty.
Then, after an maddening 90 minutes, Bohemians might have snatched it in stoppage time at the death.
Ryan McEvoy’s free-kick from the edge of the area had keeper Srdfan Rajkovic flapping before he finally gathered at the feet of in the inrushing Kevin Feely.
Though they enjoyed plenty of possession, Bohemians will simply have to be more clinical in front of goal in Akureyri next Thursday if they are to progress.
Manager Callaghan surprised no-one by fielding the same starting the same team that trounced arch-rivals Shamrock Rovers 4-0 last week.
Contrary to the confidence Bohemians brought into the game, European debutants Thor arrived on the back of two league defeats which saw them drop to fourth place in the Icelandic second division.
But with a loud, if tiny, band of travelling support behind them, Thor were well up for it with striker Sigurdur Kristjansson coming in late to upend Bohemians’ keeper Andy McNulty inside the opening minute.
Bohemians quickly settled to take the game to the Icelanders creating their first real chance on 10 minutes. McEvoy whipped a dangerous ball into the area with fellow midfielder Dave Mulcahy’s header just off target.
If shaky at the back, Thor looked threatening at the business end of the field and on 22 minutes Sveinn Jonsson surged forward from midfield to feed Ingi Hilmarsson whose low drive just hadn’t enough sting to trouble McNulty.
The ball was in Rajkovic’s net minutes later, but wasn’t to count as Peter McMahon was adjudged to have handled it before poking it home after delightful inter-play by Keith Ward and Dwayne Wilson.
Unhappy with that decision, Bohemians were glad of another handball going their way in their own box seven minutes later.
Andri Albertsson diagonal ball into the home area bounced up and clearly struck Evan McMillan on the arm. And though it was ball to hand,Bohemians were relieved to see referee Jovan Kaludjerovic wave away Thor’s appeals for a penalty.
Rajkovic then showed his nerves when, under no pressure, he needlessly punched away a long-range drive from Keith Ward in the 37th minute.
Thor then had their first real sight of goal on 56 minutes.
Bohemians failed to clear their lines and striker Johann Hannesson’s low shot was fumbled round a post by McNulty for the visitors’ first corner of the night.
McNulty was proving to be the busy keeper in the second half as he had to get down smartly to hold a drive from Hilmarsson.
Try as they might, Bohemians continued to struggle in front of goal before also snatching a valuable lead on 83 minutes.
McMillan got forward to find McEvoy and the little midfielder’s powerful left-footed drive was narrowly wide.
BOHEMIANS: McNulty; Heary, McMillan, Feely, Byrne; Buckley, Mulcahy (Traynor, 84 mins), McEvoy, Ward, McMahon (Scully, 58 mins); Wilson (Moore, 76 mins).
THÓR AKUREYRI: Rajkovic; Andri Albertsson (Bjornsson, 76 mins), Atli Albertson, Vrenko, Funicello; Jonsson, Halation (K. Hannesson, 66 mins), Hjaltalin, Hilmarsson; Kristjansson; J. Hannesson (Stroemberg, 76 mins).
Referee: Jovan Kaludjerovic (Montenegro).