It will cross the Arabian Sea to India before heading on to Myanmar, China, Hawaii and New York.
"The adventure has started," Solar Impulse chairman Bertrand Piccard said emotionally, after pilot Andre Borschberg took off from Abu Dhabi's Al-Bateen airport on the historic circumnavigation aimed at promoting green energy.
The 400 kilometre (215 nautical mile) maiden leg to the Omani capital Muscat was expected to take around 12 hours.
After more than six hours' flying, Borschberg was over the Sea of Oman with nearly 65 per cent of the distance covered.
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The takeoff by Solar Impulse 2, which was originally scheduled for Saturday but delayed due to high winds, capped 13 years of research and testing by Swiss pilots Borschberg and Piccard.
Live video streaming on the website monitoring his progress www.Solarimpulse.Com showed the pilot, wearing an orange jumpsuit, breathing into an oxygen mask.
"From Mission Control Center in #Monaco the engineers are helping me to perform Oxygen Mask tests from #SolarImpulse," he tweeted.
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 plane, 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.
Landings are also earmarked for the midwestern United States and either southern Europe or North Africa, depending on weather conditions.