Barrow, who scored 10 goals in 19 appearances for Ostersunds last season, will hope to progress to Garry Monk's senior squad after settling in at the Liberty Stadium.
"I'm delighted to be here," said Barrow, who becomes the third Swansea Under-21 squad recruit in a week following the signings of striker James Demetriou from Nottingham Forest and midfielder Giancarlo Gallifuoco, who was previously at Tottenham.
"I now can't wait to get going and improve each day in training and try to score some goals.
"I like the way Swansea play football, it reminds me of how I used to play back home. I've watched a lot of their games on TV and I was impressed with what I saw.
"They like to give young players a chance and hopefully I can progress here."
Meanwhile, Swansea teenager Alex Bray has joined Sky Bet League Two club Plymouth on loan.
The 19-year-old winger, who was part of the Swansea squad for the club's pre-season tour of the United States, links up with the Pilgrims until January 10.
Bray has yet to make a senior appearance for Swansea.