Tehran's application was backed by other founding members on Friday, China's Ministry of Finance said in a statement on its website. The United Arab Emirates' bid was also approved.
More than 50 countries, plus Taiwan, have now applied to join the bank in a diplomatic coup for Beijing after Washington initially opposed its allies becoming members.
The United States and its Asian ally Japan have not sought to join.
But US Treasury Secretary Jacob Lew said last week that Washington was "ready to welcome" the bank, though he added it should complement existing multilateral institutions such as the World Bank and International Monetary Fund.
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Iran's approval -- its application had not been previously announced -- comes immediately after the nuclear deal that China helped to broker.
Under the outline nuclear deal, the United States and the European Union are to lift all nuclear-related sanctions on Iran once the UN atomic agency has verified that Tehran has stuck to its terms.
The proposed limits will see Iran's stocks of highly enriched uranium cut by 98 percent for 15 years, while its unfinished Arak reactor will not produce weapons-grade plutonium.
There are concerns over transparency of the AIIB, which will fund infrastructure in Asia, as well as Beijing using it to push its own geopolitical and economic interests as a rising power.
Under President Xi Jinping, China is pushing to build on the ancient Silk Road trade routes on land and sea, through its "One Belt, One Road" initiative expected to be partly funded by the AIIB.