Ilves had already put together a decent run in the UECL qualifiers, beating Differdange of Luxembourg in the opening round before seeing off Iceland's Stjarnan on penalties to set up the tie with Rijeka, who had seen off Derry City with a brace of 1-0 wins.
And a third 1-0 success for the Croatian side was on the cards with just over a quarter of an hour played, as Tiago Dantas found Niko Jankovic, and from 20 yards out the midfielder powered the ball beyond Faris Krkalic and into the top corner.
Krkalic had his work cut out over the rest of the half, making seven saves and having the woodwork to thank to keep the tie alive, even at the relatively early stage.
Ilves managed to weather the storm early in the second half, and it wasn't until just before the hour mark that Krkalic was called upon again. In all, the Lynx goalkeeper made 11 stops, but his teammates failing to have a single shot in the second half meant Rijeka took the win with ease.
The Finnish side will look to make their opponents regret not putting the tie to bed when they meet again in Tampere for the second leg on 13th August.