Heavy rain at the ATP 250 clay-court in Austria on Wednesday had caused several second-round matches to be postponed until Thursday.
Further inclement weather ensued on Thursday, so the start of play delayed by 75 minutes and another stoppage following later in the day.
But not even the heavy rain could dampen Goffin's spirits, as the wildcard toppled Kohlschreiber 6-3 7-5.
The Belgian landed just 40 per cent of his first serves, but the 23-year-old fended off six of the seven break points he faced to secure a shock win.
And later on Thursday the world number 78 reached the semi-finals by defeating Paolo Lorenzi 6-2 6-1.
The result marked a mixed day for Lorenzi, who had earlier reached the last eight with a 6-2 6-0 victory over Pablo Andujar - winner of his third ATP title in Gstaad last week.
In the only other quarter-final tie of the day third seed Lukas Rosol suffered a surprise 6-3 4-6 6-3 defeat to Maximo Gonzalez - who meets Goffin for a place in the final.
In the second round, defending champion Marcel Granollers set up a quarter-final showdown with fifth seed Dominic Thiem after comfortably defeating world number 100 Diego Schwartzman 6-3 6-4, the second seed breaking serve on five occasions.
Thiem faced a slightly more testing encounter with Czech Jiri Vesely. There were five breaks in a keenly fought contest, but Thiem had enough to secure a 6-3 7-6 (7-3) success.
Andreas Seppi also progressed to the last eight, the Italian - seeded four - was granted a walkover as scheduled opponent Pere Riba withdrew with a viral infection.
Seppi now faces Argentina's Juan Monaco, whose 6-3 6-4 triumph over seventh seed Jarkko Nieminen was disrupted by the weather before play eventually reached a conclusion.