The app - called tbh, meaning "to be honest" - is just nine weeks old, but had already been downloaded 5 million times.
The app's creators said it will remain a standalone program but will now have more resources thanks to Facebook.
"We were compelled by the ways they could help us realise tbh’s vision and bring it to more people," tbh said.
According to start-up news site TechCrunch, the deal was for "less than $100m", and tbh's four person team would become Facebook employees.
In a statement, Facebook said: "tbh and Facebook share a common goal of building community and enabling people to share in ways that bring us closer together.
"We’re impressed by the way tbh is doing this by using polling and messaging, and with Facebook’s resources tbh can continue to expand and build positive experiences."
tbh said the app's success was a sign of teenagers craving more positive interactions online.
"While the last decade of the internet has been focused on open communication, the next milestone will be around meeting people’s emotional needs," it said.
The firm's achievement has been to create an anonymous app that hasn't descended into a cesspit of trolling and harassment - something many apps before it have dramatically failed to do.
After a user uploads their contacts, the app will ask pre-determined, positive questions such as "best to bring to a party?", and the option of selecting one of four friends.
Users are notified when they are selected, but the details of who chose them is kept anonymous.
Mimicking Facebook's early growth - where it was only available in a handful of colleges for a short time, the makers of tbh only made the app available to users in certain states. Word of mouth would spread at schools as the app was enabled.
"We shipped it to one school in Georgia," explained co-founder Nikita Bier, speaking to TechCrunch.
"Forty percent of the school downloaded it the first day. The next day it was in three more schools, and then the next day it was in 300 schools."
Facebook would not provide any more details about the deal, but the firm is clearly eager to snap up the next big thing in its infancy, save it become another competitor like Snapchat.
An investment bank's recent survey of 6,100 US teens suggested that Snapchat was the preferred social media platform for teenagers - the average age of participants was 16.
Facebook reportedly tried to buy Snapchat in 2013 for $3bn. Snap, the parent company of Snapchat, is today worth $19bn.
Photo by: PA Image Caption: Facebook now has over 2 billion users worldwide