The Nordic nation pledged 1.6 billion kroner ($258 million) over the next four years.
Norway had previously announced a $33-million contribution during a UN climate summit in New York last September.
"With this announcement, the total amount pledged to the fund has now reached the equivalent of $9.95 billion," the Norwegian foreign ministry said in a statement.
"Thanks to this increase in Norway's contribution, the goal of $10 billion is now within reach."
Hitting the initial goal for the fund would be seen as a key step ahead of international talks in Paris next year on slashing worldwide carbon emissions.
The South Korea-based fund aims to help developing nations invest in clean energy and green technology and build up defences against rising seas and worsening storms, floods and droughts.
It would help those countries least to blame for, but most at risk from, climate change with grants, loans and private capital for projects such as solar and wind farms, planting trees or disaster-proofing infrastructure.
"Rich countries must provide the greatest share of the funding, but all countries that have the economic capacity should contribute," Norwegian Climate and Environment Minister Tine Sundtoft said in the statement.
"Recipient countries have a particular responsibility for providing conditions that attract climate investments."
Also on Friday, the UN revealed in a report that developing countries may need as much as $250-500 billion per year by 2050 to deal with the consequences of climate change.
The estimated costs for adaptation are several times higher than in previous forecasts, according to the United nations environmental Programme (UNEP), which warned of a "significant funding gap after 2020".
"Adaptation costs could climb as high as $150 billion by 2025/2030 and $250-250 billion per year by 2050," the report said.