"@bertrandpiccard lands in #Seville completing, in 70h, the 1st #Atlantic solar flight #futureisclean," the support team's official Twitter account said.
"I can't take it in, it is so fantastic," Swiss pilot Bertrand Piccard told the plane's mission control centre in Monaco in remarks broadcast live on the Internet.
Applause broke out as the plane set down at Seville airport in southern Spain where a team was on the ground to welcome Piccard.
"A dream is coming true," the official account said as the plane made its slow approach to its final destination.
Piccard had kept followers up to date with sightings of whales and icebergs, as well as occasional turbulence on the near three-day transatlantic journey.
The voyage marks the first solo transatlantic crossing in a solar-powered airplane, and Piccard has been getting little sleep as he survives on short catnaps.
Solar Impulse is being flown on its 35,400-kilometre trip round the world in stages, with Piccard and his Swiss compatriot Andre Borschberg taking turns at the controls of the single-seat plane.
Borschberg piloted a 6,437-kilometre flight between Japan and Hawaii that lasted 118 hours, smashing the previous record for the longest uninterrupted journey in aviation history.
The plane, which has just completed 15th leg of its east-west trip, set out on March 9th, 2015 in Abu Dhabi, and has flown across Asia and the Pacific to the United States with the sun as its only source of power.
After the Atlantic crossing, Piccard and his colleague have two options: they can either make their way to Abu Dhabi with one more stop, as originally planned, or they could try to fly the rest of the way in one go.
The plane typically travels at a mere 48 kilometres per hour (36 miles per hour), although its flight speed can double when exposed to full sunlight.
Borschberg and Piccard say they want to raise awareness of renewable energy sources and technologies with their project.