Elpida has been searching for an investor to sponsor its restructuring after filing for bankruptcy protection in February with $5.6 billion in debt.
Micron also offered to keep Elpida's two main factories in Japan open and to guarantee jobs for the company's current employees.
It outbid Hynix, which dropped out of the race on Friday.
According to Reuters, Micron's offer was accepted by Elpida's trustees and management.
A final restructuring plan will still require the approval of a local court and Elpida's creditors who might not have been happy at a low bid for the company.
The Japanese government will be happy to see the back of the failed company. It had propped up Elpida with public funds to save what was billed as the country's last hope for the DRAM market.
Elpida ranks third in the DRAM market, but analysts think that in the long term Elpida DRAM chips will do well as mobile device demand increases.