The flight route has taken Konyukhov from Australia to above New Zealand, across the Pacific Ocean, South America, the Cape of Good Hope and the Southern Ocean.
Winds pushed his flight towards Antarctica as he made his approach to Australia on the final leg of the journey but he successfully crossed over land in the country's west early on Saturday.
Wallington said the last 24 hours of the trip had been uneventful for the experienced Russian explorer but joked that "the previous 10 days have been awful."
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He said flying through thunderstorms and getting pushed towards Antarctica by strong winds had been particularly hard.
Konyukhov, who has previously conquered both the north and south poles solo, and set a record sailing a 27-meter-long (89-feet) boat round the world alone, had not been motivated by breaking the record, Wallington said.
"He just thought it would be a really nice thing to do," he said of the round-the-world journey.
"Breaking a record is a nice bonus, but the objective was just to fly around the world."