Jung won aboard Sam, who was a late replacement when Fischertakinou contracted an infection in July.
France's Astier Nicolas took silver and American Phillip Dutton won bronze as GB's William Fox-Pitt finished 12th.
Fox-Pitt, 47, was competing just 10 months after being placed in an induced coma following a fall.
The three-time Olympic medallist, aboard Chilli Morning, produced a faultless performance in Tuesday's final show-jumping phase.
But the former world number one's medal bid had already been ruined by a disappointing performance in the cross-country phase on Monday.
He led after the dressage phase, but dropped to 21st after incurring several time penalties in the cross country.
Ireland's Jonty Evans finished ninth on Cooley Rorkes Drift.
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The bird, known officially as LF15, and her partner LM12 bred three chicks in both 2015 and 2016 at the Dunkeld reserve.
Lassie arrived back in Scotland on 23 March, six days after her mate.
Viewers from around the world have been watching the pair's progress on the Loch of the Lowes live webcam.
Charlotte Fleming, Perthshire ranger for the Scottish Wildlife Trust, said: "LF15 has settled down to begin incubating, and we would expect the egg to hatch in about six weeks' time.
"It's now clear that the pair's attempts at breeding have been a success and they are on track to repeat the hat-trick of chicks that were hatched in both 2015 and 2016."

Research on 1,500 women aged 64 to 95 found those who spent many hours sitting and exercised for less than 40 minutes a day had cells that were biologically eight years older.
As people age, their cells age, causing DNA protectors to shorten and fray.
But health and lifestyle factors may speed up the process, researchers from California said.
Even in old age, it was important to keep active and avoid sitting for more than 10 hours a day, they said.
During ageing, tiny caps on the ends of DNA strands naturally shorten.
These telomeres - which have been likened to the plastic tips of shoelaces - are there to prevent chromosomes from deteriorating.
Telomere length is one indicator of biological age, which does not always match chronological age.
Shortened telomeres have been linked with an increased risk of cardiovascular disease, diabetes and major cancers.
How much regular exercise people do now also appears to be a factor in their length.
To track the women's movements in the study, they wore wore an accelerometer on their right hip for seven days in a row, during the day and night.
They were also asked to complete questionnaires on their activity.
Dr Aladdin Shadyab, lead study author, from the University of California San Diego School of Medicine, said: "We found that women who sat for longer did not have shorter telomere length if they exercised for at least 30 minutes a day.
"Discussions about the benefits of exercise should start when we are young, and physical activity should continue to be part of our daily lives as we get older, even at 80 years old."
The study is published in the American Journal of Epidemiology.
NHS Choices advises older adults to break up long periods of sitting with light activity because "sedentary behaviour is now considered an independent risk factor for ill health".
It says adults aged 65 or older who are generally fit and mobile should try to do:
Older adults at risk of falls, or with poor balance, should also do exercises to improve balance and co-ordination at least twice a week.
Germany's Michael Jung retained his Olympic individual eventing title by winning gold at Rio 2016 on his second-choice horse.
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A female osprey nicknamed Lassie has laid her first egg of the season at the Scottish Wildlife Trust's Loch of the Lowes nature reserve.
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Women who lead a sedentary lifestyle have faster-ageing cells than those who exercise every day, research suggests.