The 21-year-old has put pen to paper on a four-year contract, keeping him at Parkhead until 2025.
His previous deal ran until 2023.
Having made his debut against Hamilton in February 2020, this season has been a breakthrough campaign at Celtic having played 18 games during a loan spell with Greenock Morton in the Championship.
Welsh has made 16 appearances in all competitions and has started the last seven games, helping the team keep five clean sheets.
“I’m delighted to get it done and I’m here for another four years which is the perfect scenario for me,” Welsh told Celtic TV.
“I’ve been getting a consistent run of regular games since the Christmas period there. For me, it’s just about keeping that up, keep doing it, keep playing well until the end of the season.”