The aircraft touched down in Muscat after nightfall, 13 hours and two minutes after taking off from Abu Dhabi.
Pilot Andre Borschberg, who was at the controls on the 400-kilometre trip, smiled and waved to his team after landing.
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The takeoff by Solar Impulse 2, which had originally been scheduled for Saturday but was delayed because of high winds, capped 13 years of research and testing by Swiss pilots Borschberg and Piccard.
Live video streaming on the www.Solarimpulse.com website monitoring the unique aircraft's progress showed the pilot, wearing an orange jumpsuit, breathing using an oxygen mask.
"From Mission Control Center in #Monaco the engineers are helping me to perform Oxygen Mask tests from #SolarImpulse," he tweeted.
Borschberg earlier called his wife from on board, according to the live feed.
Shortly before takeoff, the 63-year-old pilot tweeted that the "challenge to come is real for me & the airplane".
The wingspan of the one-seater known as the Si2 is slightly bigger than that of a jumbo jet, but its weight is around that of a family car.
From Muscat, it will make 12 stops on an epic journey spread over five months, with a total flight time of around 25 days.
On Tuesday, it is expected to cross the Arabian Sea to Ahmedabad in India before later legs to Myanmar, China, Hawaii and New York.
Landings are also earmarked for the midwestern United States and either southern Europe or North Africa, depending on the weather conditions.
The longest single leg will see a lone pilot fly non-stop for five days and nights across the Pacific Ocean between Nanjing, China and Hawaii, a distance of 8,500 kilometres.
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