"It's a dream but we did believe it could happen," Geale told Fox Sports after an impressive performance in Oberhausen.
"We thought we could go over there and do it. He's been a great fighter for many years but he'd never fought me before. We had a game plan and stuck to it pretty well.
"There was a couple of times where I caught him with some power shots and hurt him," added the Tasmanian-born Geale, who fights out of Sydney.
British official Dave Parris and South African Stanley Christodoulou both scored the fight 116-112 in Geale's favour, while American judge Eugene Grant had Sturm ahead 116-112.
Geale began to take control of the fight in the middle rounds, hurting Sturm with some stinging body blows, but the German hit back late, making it hard for the judges.
"I'm feeling a little bit tired but pretty good," said the 31-year-old Geale, who became the first Australian-born boxer to unify two globally recognised titles in the same division.
"I'm still extremely hungry and want to win more, beat more champions," he added after a successful third defence of his IBF title improved his record to 28-1, with 15 knockouts.
"The middleweight division has got lots of strong fighters and I beat one of them tonight. Now I want to take on some of the others."
Sturm, unbeaten in six years coming into the fight, left the ring with a record of 37-3-2, 16 of his wins coming by KO.