Poor weather last weekend prevented Ms Oh from reaching the summit of Annapurna, at 26,545 feet (8,091 metres) - a notoriously difficult peak to climb.
Her nearest rival in the 14-peaks quest, Spanish climber Edurne Pasaban, has expressed scepticism about her claim to have reached the summit of Kangchenjunga in 2009.
Ms Oh has denied all the allegations. Her sponsors called a press conference in December 2009 to reassert her claim to have reached the summit.
All climbers who make an ascent from Nepal report to Elizabeth Hawley, an 86-year-old American based in Kathmandu, whose research is recorded in the Himalayan Database.
They have to answer her questions about the climb they have just undertaken.
She is widely accepted as the arbiter of Himalayan climbs. There is no official body that authenticates claims.
John Sudworth, BBC News, Seoul
The South Korean public will want to give Oh Eun-sun the benefit of the doubt. Hers is a story that has much popular appeal, a plucky, tenacious climber who came from behind to beat those who once seemed much better positioned to win the prize.
Her record attempt has certainly sparked a media frenzy, today's climb was broadcast live on national television with the help of a camera carried by another professional mountaineer who accompanied Ms Oh to the top. Advertisers have also been getting in on the act, with companies buying space in this morning's newspapers to wish her luck.
But her success has also been overshadowed by the news that, just hours before Ms Oh began her attempt, two other Korean mountain climbers were missing on the same mountain, and four others hospitalised.
But Ms Hawley has marked Ms Oh's 2009 climb as "disputed" and says that Ms Oh and her Sherpa will have to be questioned again about that ascent when they return from Annapurna.
One of the key issues at the heart of the dispute is a still photograph that, Ms Oh's sponsors say, was taken at the summit but not on the "three or four square feet" right at the top.
Miss Hawley says the photograph is "clearly" not taken at the summit, because Ms Oh is standing on rocks and "summit pictures of other people on the same mountain in the same season show them standing in the snow". However, pictures taken by climbers in previous years have shown rocks close to the summit.
Ms Oh's ascent of Kangchenjunga remains recognised in the Himalayan Database, so she can claim the record.
However, if Ms Hawley's further investigations lead her to change the status of the 2009 ascent to "unrecognised" Ms Oh would not be internationally regarded as the first woman to have climbed all 14 8,000ers, as they are known.
Twenty men have scaled all 14 summits - the latest climber completed the quest just 75 minutes before Ms Oh.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.