The 25-year-old joins for an undisclosed fee, in the region of €4m, on a four-year contract.
Benson underwent a medical yesterday, after the clubs came to an agreement over the player on Tuesday.
And, having signed before the noon cut off point on Friday, he will be available to face Luton Town on Saturday in Burnley’s first home game of the season.
When asked about what Benson will bring to the side, Kompany said: “It is the business end of the pitch and these type of things are sometimes a little bit harder to coach.
"Someone like Benson is a player I believe still has a lot of progression in him.
"I think he is a player who will give us that little bit of instinctive performance in that final third.”