The title, which has sold over 54 million copies, allows players to build structures with retro Lego-style blocks, as well as explore a large map and battle others.
The deal was announced by Xbox chief Phil Spencer.
Mojang, whose three founders will leave the company, assured fans that "everything is going to be OK".
Some analysts have speculated the deal is designed to attract more users to Microsoft's Windows Phone devices.
The acquisition comes a year after Microsoft bought the handset and devices division of Finnish mobile phone firm Nokia.Minecraft creator Markus Persson has been critical of Microsoft
Minecraft is one of the top-selling apps on both Apple's iOS store and and Android's Google Play, and has recently been released for the Xbox One and PlayStation 4, further boosting sales.
Last month, it was the third most popular console game, according to market research firm NPD Group, despite being on sale for a while.
The game's developer, Mojang, which was founded in 2009, brought in over $100m in profit last year, and employs about 40 people.
Microsoft said the Mojang team would join its game studio, which is responsible for titles such as Halo, Forza and Fable.
The tech giant's chief executive, Satya Nadella, said: "Minecraft is more than a great game franchise - it is an open world platform, driven by a vibrant community we care deeply about, and rich with new opportunities for that community and for Microsoft."Opposition to sale
Mojang's founder, Markus "Notch" Persson, has previously criticised Microsoft, and commented to Reuters that the market for Windows phones was "tiny" and not worth developing apps for.Minecraft encourages users to build their own versions of the game, such as this Hunger Games title.
Minecraft has a large and enthusiastic cult following, many of whom have reacted angrily to what they see as a corporate takeover of a communally-spirited independent company.Continue reading the main story “Start Quote
Not only is [Minecraft] profitable, but it continues to increase in profits years after its release”End Quote James McQuivey Analyst, Forrester
"Makes me sick, and sad," wrote one user on a popular Minecraft forum. "It would kill the gaming community," EvilBatsu added.
Others expressed concerns about whether fans would be able to exhibit their skills.
"Not only will it cost more money to play the game it will cost people their jobs too. Many people play Minecraft and upload it to YouTube as their career, but if Microsoft takes over there will for sure be copyright issues."
However some enthusiasts made the point that Microsoft could devote larger resources to upgrading and expanding the game.
In a statement, Microsoft said it would maintain Minecraft across all its existing platforms, with a "commitment to nurture and grow it long into the future".
It added that the acquisition was expected to be concluded by the end of 2014.
In an announcement confirming the deal on its website, Mojang reassured gamers, saying: "Please remember that the future of Minecraft and you - the community - are extremely important to everyone involved. If you take one thing away from this post, let it be that."
With regard to Microsoft, Mojang said: "There are only a handful of potential buyers with the resources to grow Minecraft on a scale that it deserves."
The firm added that "Notch" had decided that he "doesn't want the responsibility of owning a company of such global significance".
Developer Notch on selling Minecraft:
"I've become a symbol. I don't want to be a symbol, responsible for something huge that I don't understand, that I don't want to work on, that keeps coming back to me. I'm not an entrepreneur. I'm not a CEO."
Read his statement in fullWhy Minecraft?
James McQuivey of analytics firm Forrester, noted that "Minecraft is one of the most important gaming properties in the world".
"Not only is it profitable, but it continues to increase in profits years after its release, largely due to the passionate fan base that invests in building out their own Minecraft worlds.Many Minecraft enthusiasts build detailed structures, such as this house
"That helps explain why Microsoft would want Minecraft and would want to ensure it is always available on Microsoft's gaming platforms.
Prof Mark Skilton, from Warwick Business School, said the acquisition was about building a "strong customer base" for Microsoft.
"The online gaming industry is fast moving from niche collective enthusiast to mass market and Minecraft is a logical move as big business follows the traffic numbers in the digital world."
Last month, Amazon bought Twitch, a site which allows users to watch other people play video games, for $970m (?597m).