The 62-year-old led his native South Africa to World Cup glory in 2007 and coached teams in Australia, France and Japan before being handed the Fiji job this month.
He made a winning start in a roller-coaster game in the rain against a Japan side coached by Jones.
The Australian Jones was White's technical adviser for South Africa's World Cup win.
White picked no Europe-based players for the game and handed out four new caps.
He said Fiji "can beat anyone" on their day and backed them to make an impact at the World Cup in Australia in a year's time.
"If we can just give them a little bit of control and structure and a bit of organisation, and I can get the best 33 players together for the World Cup, I genuinely believe that they can be a massive force at the World Cup," he said.
Japan and Fiji will both head to Europe for next month's Nations Championship games.
Japan will play Wales, England and Scotland, while Fiji will take on France, Ireland and Italy.
White will have his Europe-based personnel available for those games but he was impressed by the players he picked to face Japan.
"If that's the calibre of player that we can put together from the island, there's no reason why they can't be a force at the World Cup," he said.
Both teams scored three tries in the first half and one more each after the break.
Fiji substitute Ben Volavola made the difference with two penalties late in the game.
The two sides had met in the Pacific Nations Cup final in Tokyo a month earlier, with Japan winning 20-15.
The rematch was being played to mark the 100th anniversary of the Japan Rugby Football Union.
"We know that Fiji can play with flair, we know they've got unbelievable skill -- all I wanted to do was bring a little bit of control," said White.
"The way we played in the last 15 minutes really made me proud."
Naoto Saito gave Japan an early lead before tries from Kitione Salawa and captain Tevita Ikanivere put Fiji in front.
Japan hit back through captain Warner Dearns before Fiji regained the lead through Tuidraki Samusamuvodre.
Faulua Makisi brought Japan level going into half-time but Terio Tamani scored for Fiji minutes after the restart.
Japan struck back with another Dearns try but Volavola stayed cool to edge Fiji back ahead for good.
Jones said the "biggest difference in the game was the number of turnovers".
He also backed White to make an impact with Fiji before the World Cup.
"Today he brought in four or five new caps and I saw him talking about going out to all the islands in Fiji to find more talent," said former Australia and England coach Jones.
"The one thing he'll do is find the best talent in Fiji and bring that forward."
Nations Championship 2026
The inaugural Nations Championship will see six northern hemisphere and six southern hemisphere nations battle it out across six rounds in the July and November international windows before assembling in Twickenham for one unforgettable finals weekend on November 27-29.
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