Chelsea's sister club had the upper hand for most of the match, but thanks to a strong performance from Patrick Pentz and a disciplined effort from the Danish visitors, it ended goalless at the Stade de la Meinau.
The return leg at Brondby Stadium next Thursday will therefore decide who gets a place in the Conference League.
The match had barely started before it was interrupted again. Heavy smoke from the Brondby's fan section meant that visibility was virtually non-existent in the arena, which was far from full due to extensive renovations.
The smoke lifted, and the young Strasbourg team without the injured Lucas Hogsberg grabbed the initiative as expected, without it translating into goalscoring chances.
Brondby were content defending deep and relied on counter-attacking opportunities where Nicolai Vallys twice got the chance to put the visitors ahead but failed to keep his cool at the decisive moments.
Strasbourg also came close to making life miserable for themselves with ten minutes to go before half-time, when Valentín Barco curiously carried the ball past his own goalkeeper and then kicked it away on the goal line.
The pattern continued after the break, with Strasbourg launching a virtual siege on the Brondby goal.
Pentz was forced to make a couple of miraculous saves from close range attempts, but the goalkeeper was at his mercy with 20 minutes remaining, when Joaquin Panichelli, in a completely free position, hammered the ball far over the goal at the back post.
Time worked in Brondby's favor, but Strasbourg came close to the winning goal in the final minutes, when Dilane Bakwa was denied by the woodwork.
In added time Pentz showed his sublime reflexes again from Emanuel Emegha's shot from close range to give the Danish visitors a vital goalless draw before the return leg.