In the first leg, Brondby keeper Patrick Pentz shone with several crucial saves, but in the second leg, he couldn't live up to his performance in Strasbourg.
Big mistakes by the Austrian international gave the visitors two goals in the first half, while Daniel Wass reduced the deficit shortly before the break to ensure a dramatic second half.
Strasbourg captain Emmanuel Emegha punished an otherwise bravely fighting home team on a counterattack and increased the lead to 3-1 midway through the second half.
However, the final minutes still proved nerve-wracking when newly acquired attacking midfielder Michael Gregoritsch converted an indirect free kick from close range.
However, Strasbourg stood firm to survive the Danish siege towards the end, and with Brondby's exit, Denmark will be represented in the autumn by FC Copenhagen in the Champions League and by FC Midtjylland in the Europa League.
Brondby has not won a decisive qualification tie and advanced to a group or league phase in 20 years now.
Current national team player Lucas Hogsberg played the entire match for Strasbourg after his transfer from FC Nordsjaelland and put in a strong performance before the World Cup qualifiers against Scotland and Greece.