The 30-year-old, who has won 19 caps for Argentina but missed out on their World Cup squad, has agreed a one-year contract with an option for a further 12 months.
Bernardo scored 18 tries in 42 appearances for Bristol, and Glasgow are convinced they have signed a first-class number 15, following the departures of Rory Lamont to Sale and Francisco Leonelli to Saracens.
Warriors head coach Sean Lineen said: "I was looking to add some experience to the backline and losing Rory Lamont and Francisco Leonelli also left us short.
"Having looked at some footage and spoken to Richard Hill at Bristol and Argentina's director of rugby, Les Cusworth, I know Bernie will bring a lot of energy and experience to the Warriors' backline."
Stortoni said: "The boys are all very friendly and they know my English isn't very good, so I must say thank you to them for taking the time to teach me all the different moves."