Although the new system is voluntary, 1,207 of the 1,340 who have so far been offered the tests have agreed to undergo them.
The agency said that 470 people had already had the tests carried out, with the first results expected in the next month.
"I'm very confident that the system that we have developed, that this is a process that ensures asylum seekers' rights," Elias Palm, deputy director general of Sweden's Department of Forensic Medicine, told Radio Sweden.
The risk of us determining a child as an adult is a lot less than the risk of determining a 19 to 20-year-old as a child. But then again, when you are 22 or 25, the risk of you being determined as a child is extremely small."
The new system combines magnetic imaging of asylum seekers' knees with X-rays of their wisdom teeth.
Palm and his colleagues chose this method because knees and wisdom teeth on average reach maturity at age 20 or even 22. It is very rare for the wisdom teeth and knees of those under 18 to be fully developed.
Over 35,000 unaccompanied child migrants came to Sweden in 2015, more than came to any other country in Europe.
According to Åsa Carlander Hemingway, unit head for the Swedish Migration Agency, only 12 percent of the child migrants had any sort of paper ID. Among the Afghans, it was just two percent.
Carlander Hemingway said that adults who successfully claimed to be children were rewarded with full access to healthcare and education, a lawyer, and arguably better accomodation.
"A lot of people also think that they have a better chance of gaining asylum if they claim to be children," she said. "These are all factors that make it interesting to claim that you're under age."
The organisations carrying out the tests – Boneprox and the Karolinska Institute for teeth and Unilabs and Aleris for the knees – are capable of handling about 500 tests a month.
The Migration Agency expects that between 3,000 and 14,000 tests will be carried out each year.
When plans to carry out medical age assessments were announced at the start of 2016, many doctors said they would refuse to carry them out as no existing test was sufficiently accurate to fairly determine age.