Pasaban, 36, reached the top of the 8,091-metre mountain along with several other Spanish climbers at around 2 pm (1430 IST), a spokesman for her team told AFP.
She now has just the 8,027-metre Shisha Pangma to scale in her bid to be the first female to reach the summit of every mountain over 8,000 metres (26,247 feet).
But her chief rival, South Korea's Oh Eun-Sun, is also on the slopes of Annapurna, which would be her 14th and last summit.
Annapurna is particularly dangerous because it is both technically difficult and avalanche-prone, and it has a much higher death rate than Everest, the world's highest peak.
Pasaban had been defeated by Annapurna once before, in 2007 when she and her team turned back about 1,000 metres from the summit in bad weather.
She will travel to Tibet to attempt her scale of Shisha Pangma "over the coming days", her spokesman said.
A third contender, Austria's Gerlinde Kaltenbrunner, is to begin her ascent of Mount Everest, the world's highest mountain, next week.
But she would still have to climb K2, the world's second highest peak situated on the Pakistan-China border and regarded as the most difficult and dangerous of the 14 'eight-thousanders', in order to claim the record.
Italy's Reinhold Messner became the first man to climb all 14 summits in 1986.
The 14 mountains are all located in the Himalayan and Karakoram ranges in Asia.
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