The 19-year-old has agreed a three-year deal at the end of his City contract, and joins Scott Twine and Luke McNally through the door this week.
The Manchester-born teenager can operate as a right back, centre back and a holding midfielder, and he joins the Clarets after 12 years at the Etihad Stadium, where he progressed through the academy ranks, captaining both the Under 18s and Under 23s.
Kompany said: “CJ is a player we have been watching closely for a long time.
"He is a promising and talented footballer with an excellent mindset and attitude.
“We are delighted he has agreed to join. CJ has shown a real hunger and he wants to learn and succeed at Burnley.
"He fits perfectly into our culture and he is a fantastic addition to our squad.”
After signing at Turf Moor, Egan-Riley said: “I’m buzzing to be here; happy the deal is over the line and I’m a Burnley player.
“I just can’t wait to get pre-season started and get the season underway.”