The Hydra, created by patients of the Craiglockhart War Hospital, included works by Owen and his literary mentor and fellow patient Siegfried Sassoon.
The recovered edition contains general notices, criticisms about hospital conditions and, most notably, poetry about life during the First World War.
The original magazines were lost over time, which led to an appeal by war memorabilia experts from Napier University.
Staff were stunned when relatives of another former patient, who took over as Hydra editor when Owen returned to the front, announced they had three copies and donated them to Napier.
The magazines will now go on display at the War Poets Collection exhibition later this year.
Librarian Catherine Walker, who curates and maintains the exhibition, said: "Having original copies back in the building where the magazines were written is a thrill and fitting as we approach the centenary of Britain going to war.
"The Hydras allow us glimpses of the daily lives of the men who found them- selves sent to Craiglockhart. We are truly surprised and delighted that they have been donated to us."
The hunt for edition number six of the series, dated July 7, 1917, is ongoing.