The 31-year-old has agreed a two-year contract and will link up with the Guinness Premiership play-off contenders this summer.
Beveridge has won six caps, making his Test debut on Scotland's tour of New Zealand in 2000, in addition to making 14 Scotland A appearances.
Warriors coach Sean Lineen admitted Beveridge's decision came as a blow, particularly as Glasgow will be better funded by the Scottish Rugby Union next year, with Borders being disbanded.
Lineen said: "Graeme has been a great servant to Glasgow and to Scottish rugby. He has been invaluable to us this season with his knowledge and leadership skills. I know the boys will be sad not to hear his chat next season.
"I've known Graeme for a long time, back to my days at Boroughmuir. My last season as a player was his first season - he talked a lot then too.
"I'll be sorry to see him go after this week's news that we will be allocated extra funding to strengthen the Glasgow Warriors squad, but it's a great opportunity for Graeme at this stage in his career."
Bristol head coach Richard Hill has moved decisively in the transfer market, given the possibility of losing Shaun Perry to England World Cup duty early next season, and with Perry's fellow number nine Brian O'Riordan sidelined for six months due to a dislocated shoulder.
Hill said: "Graeme is a fine player who has represented Scotland in the past, including being part of the 2003 World Cup squad, as well as possessing plenty of experience at Heineken Cup level.
"We could be missing Shaun for large chunks of the season, and Brian will obviously miss pre-season because of his shoulder surgery.
"So, we were looking at going into 2007-2008 with just Haydn Thomas [a recent signing from Gloucester] and, had anything happened to him, we would have been in trouble."