The teenager, who made one senior appearance for Spurs, has signed a one-year deal with Kevin MacDonald's men which includes the option of a further 12 months.
Barthram, who can play in either full-back position or in the middle of the field, had been a regular for Spurs' reserves before being released earlier this year.
"Swindon is a big chance for me and I am pleased to have everything sorted," Barthram told the club's official website.
"It's an exciting time. The training sessions have been good to far and I am looking forward to getting down to the season and playing games."