The 33-year-old, riding his 2012 London Olympic winner La Biosthetique-Sam, completed a consecutive triple of Burghley, Kentucky and Badminton that had been achieved previously only by Britain's Pippa Funnell in 2003.
Badminton, held in the grounds of the English stately home that gave its name to the racquet sport played there in the mid-1800s, is the most celebrated equestrian three-day eventing classic, featuring dressage, cross-country and showjumping.
Though Badminton is now in its 67th edition, the Grand Slam was not introduced until 1999.
Jung, a favourite to defend individual and team titles at the Rio Olympics in August, led all three days and clinched victory with a flawless round of showjumping on Sunday for a record low of 34.4 penalty points.
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"I know him from many years, from many tough competitions. He's a very experienced horse now," Jung had said after Saturday's cross-country.
"He always trusts me, listens to me and it was very nice to ride him...he gives me a very good feeling, very proud of him.
"We have an amazing partnership, so for me he is the best horse in the world."
Compatriot Andreas Ostholt finished second on So Is Et, emphasising Germany's team strength, with Gemma Tattersall improving her claims for selection to the four-strong British Olympic team with third place on Arctic Soul.
New Zealand's double Olympic gold medallist and six-times Olympian Mark Todd, still a contender at 60 years old, was fourth on Leonidas II -- 36 years on from his first success at Badminton in 1980.
New Zealand had four finishers in the top 10.
Zara Tindall, daughter of Britain's Princess Anne and an Olympic silver medallist in 2012, finished 23rd on High Kingdom.
(Reporting by Alan Baldwin, editing by Ian Chadband)
