Soliman was knocked down in round two and would have been well behind after three rounds.
However, 39-year-old Soliman produced one of the great performances of his 16-year professional career, dominating most of the rest of the fight with his superior work rate.
Sturm, who lost his WBA middleweight Super title to Geale last year, couldn't match the industry of the busy Australian, who was relentless through the second half of the bout.
Two judges scored it 114-113 to Soliman and their colleague 116-111 to the Australian.
Geale retained his IBF title on Wednesday by outpointing Anthony Mundine over 12 rounds in Sydney.
Geale's promoter Gary Shaw earlier in the week indicated he might petition the IBF to have a voluntary defence next, rather than fight the winner of Soliman-Sturm.