Sweden's Migration Board has described the influx as “extraordinary”.
“These past few weeks have seen an incredible rise in the number of unaccompanied minors,” spokeswoman Sophia Ohvall Lundberg told AFP. “The situation is strained.”
She added: “No one is saying we're not going to make it. Each player is trying to improve the system.”
Sweden's system for taking these children in may be under pressure, but so far housing has been found for each minor, and every case is being followed up by authorities.
There are many reasons why Sweden is the country of choice for these children, most of whom have escaped conflict and misery in Somalia, Afghanistan, Syria and Eritrea.
But the main explanation is that Sweden is known for its generous asylum policy and welfare state — and word of mouth travels quickly.
“They know they will be welcomed humanely in Sweden, that things will be pretty well-organised and that they have a good chance of getting asylum,” Red Cross official Ida Holmgren said.
When the children arrive in Sweden, they are registered and housed either in a special asylum centre for minors, or placed with distant relatives or a host family, which is automatically the case if they're under 13.
Social services then track the children's well-being and schooling.
The system isn't perfect. A recent report by the Stockholm's Stadsmissionen volunteer organisation said a lack of personnel was hurting children's welfare.
Many of them are traumatised by what they've been through, wary of strangers, and unaware that they may need psychological help.
Mahmud Zaraa, 13, appears unusually calm and well-adjusted on this autumn day in a cafe in peaceful Eskilstuna, chatting happily about playing football as he sips orange juice.
His manner reveals no trace of his dramatic three-month journey to Sweden, covering part of Europe on foot before ultimately ending up in this town of 100,000 people.
Mahmud is now living with a great-uncle and his family.
One day “before the summer”, he recalls vaguely, his entire family piled into two cars to leave Damascus for Europe.
Mahmud wanted to play with his 20-something cousin's mobile phone, so he jumped into that car instead of the one carrying the rest of his immediate family.
At the Turkish border, a burst of gunfire made his parents' car turn back to Syria, while the car carrying him and his two cousins forged ahead.
Mahmud now goes to Swedish class every morning, one of many programmes to help unaccompanied minors learn about the culture and language of their new home.
“Swedish is not so hard,” he insists, excited that his teachers mix games with learning, an approach to school he'd never seen before.
He hopes to join a special class for foreign students at a “normal” school in a month, and so far, he says, he's happy in Sweden.
Once he has obtained a residency permit, which can take up to eight months, the rest of his family hopes to join him under family reunification rules.
“It's very important that the children resume contact with their family, if they want to. Behind each of these faces is a horrific story,” says the Red Cross's Holmgren.
Some unaccompanied minors become separated from their families along their journey, while others are sent off alone to Sweden. Family reunifications can sometimes take years.
Mahmud is full of confidence that he will have a good life in Sweden, blissfully unaware of many of the hurdles immigrants face here, including acute housing shortages and growing anti-immigrant sentiment with a far-right party on the rise.
In 2014, the unemployment rate for people born outside Sweden was double the national average.
One of Mahmud's distant cousins in Sweden, Bashar, a 28-year-old lawyer who fled Damascus two years ago, is still struggling to learn Swedish.
He fears he'll never be able to practice law here but is not giving up yet, hoping to go back to school to study international law.
“I have experience but my diplomas have to be translated and to be a lawyer you really have to master the language,” he says.
In the meantime, he's just grateful that Sweden has given him a chance to rebuild his life — and Mahmud the chance to grow up in peace.