The 21-year-old has agreed a three-year contract at Oakwell after making 42 appearances for the Scottish Premiership club last season.
With six goals for the Firhill side last term, Barnsley's third signing of the summer was the only player outside of Aberdeen and Celtic to make the Premiership Team of the Year.
Lindsay told the official Tykes website: "I'm delighted to be here. I met the gaffer (Paul Heckingbottom), he took me into his office and sat down with me outlining his plans for the team, for me and for the club.
"I was really impressed with that. It made me feel good knowing he knew what I was about, that he knew my game. Everything was so detailed, it's a really professional setup and I can't wait to get started."
The centre-back was reported to be on the verge of a move to Oxford United earlier in June, but the transfer allegedly fell through due to uncertainty over the position of U's manager Michael Appleton, who has since left for Leicester City.