Rijeka had broken new ground by reaching this stage of a UEFA competition for the first time in their history, and the home side would have wanted to keep things tight in the opening minutes while their nerves settled.
However, Strasbourg - who finished top of the league phase - had other ideas, taking the lead after just 73 seconds.
A swift foray forward culminated with Sebastian Nanasi cutting inside from the left flank before picking out Joaquin Panichelli, who took a touch and fired past Martin Zlomislic for just his second goal of the UECL campaign.
The Argentine striker could have - and probably should have - added to that tally, but instead missed a host of chances.
First, he tried to lob Zlomisilic, yet his effort sailed wide, before the Rijeka goalkeeper made a save down to his right from a powerful shot just inside the box.
Panichelli remained his side’s biggest threat as the half progressed, although his frustrations grew as he was denied by the woodwork.
The Croatian side were largely second-best throughout the first half but did have the opportunity to draw level midway through the first period via Toni Fruk. His shot from a narrow angle, though, was well saved by Mike Penders.
As in the first half, Strasbourg were quick out of the traps following the restart in their pursuit of the second goal. Julio Enciso fashioned a chance five minutes after the interval, powering into the box and releasing a powerful shot, only to be denied by Zlomisilic.
Andrew Omobamidele was next to try his luck, but his effort sailed over the bar. The away side continued to probe and did have the ball in the back of the net just after the hour mark after Nanasi curled in an excellent strike - on this occasion, though, the offside flag denied him.
Le Racing did manage to increase their lead with a quarter-hour remaining when Martial Godo - who came on for Nanasi - fired home a rebound from close range after Guela Doue’s initial strike was saved.
Still, O’Neil was left frustrated when Rijeka pulled a goal back almost immediately to halve the deficit. Tiago Dantas’ free-kick into the box was powered home by Daniel Majstorovic, who was left unmarked in the box.
While Rijeka went in pursuit of a second goal, they struggled to break down the Strasbourg rearguard, leaving the tie finely poised ahead of next week’s second leg.
The French side find themselves in control and will be confident they can get the job done on home turf, but Rijeka were unbeaten on their travels in the league phase, and so they will back themselves to turn the tie around.
Flashscore Man of the Match: Joaquin Panichelli (Strasbourg)
