The law, which will come into force on July 1 next year, has been met with resistance by some electronics retailers, who say the legislation was adopted without consulting them.
The law has been presented as a way to help Russian IT firms compete with foreign companies and spare consumers from having to download software upon purchasing a new device.
The country's mobile phone market is dominated by foreign companies including Apple, Samsung and Huawei. The legislation signed by Putin said the government would come up with a list of Russian applications that would need to be installed on the different devices.
Russia has introduced tougher internet laws in recent years, requiring search engines to delete some search results, messaging services to share encryption keys with security services and social networks to store user data on servers in the country.