Tries from Mauro Bergamasco, Andrea Scanavacca and Kaine Robertson in the first six minutes handed Pierre Berbizier's side an astonishing 21-0 lead at Murrayfield.
Scotland hit back with tries from Rob Dewey and captain Chris Paterson but a late try from Alessandro Troncon made sure of the historic win for Italy.
The closest the Azzurri had previously come to a Six Nations win away from Rome was in Wales last year when they secured a well-deserved draw.
They went one better in Edinburgh to signal that they are now truly comfortable in the Six Nations - and will be a threat to Scotland in the World Cup in September.
The teams meet in seven months' time in what is almost certain to be the decisive encounter in World Cup Pool C and in St Etienne Scotland coach Frank Hadden will hope his side do not reproduce the early errors which cost his side so dearly in Edinburgh.