The 22-year-old has signed a three-year contract, and will make his Clarets bow in Saturday's final pre-season friendly at home to Inverness.
Coyle feels the youngster has "an unbelievable career" ahead of him, and expressed his delight: "I'd like to go on record to thank Sir Alex Ferguson and Manchester United for their help.
"To get a player of this stature on board I think it shows what we're trying to do and what we've spoken about at this football club, and to entice a player of his quality, 22 years of age and, I think an unbelievable career ahead of him, we're just delighted to have him on board."
Eagles comes after netting twice for United in Africa against Kaizer Chiefs and Portsmouth, and Coyle added: "With Chris away with Manchester United, Sir Alex has trusted him to play the amount of games he played, so it shows the quality he's got to play with that level of player.
"He's shown that over the years.
"The one thing Sir Alex has said is that we're getting a quality player, a hungry player and he's shown he can play at that level very comfortably. So we're excited by that.
"I think our players will love playing with him, and I think equally he'll enjoy playing with the company he's playing with now."
And Coyle sees Eagles playing in an advanced position: "He can play anywhere but I see him playing in a very attacking role for us and looking to get forward at every opportunity. He's a creative player, he scores goals, he's capable of taking players on.
"He just has an array of talent and we're just delighted to get him at the football club."
Eagles admits there was only one destination for him after a meeting with Coyle, having been impressed with the club's ambition: "At the end of last season I was thinking about (my future) because my chances had been very limited at United.
"I met up with the gaffer just for a chat because I didn't know what I was going to do. I've been at United for seven years, since I was 14, so it's been long. But when I met him I was so impressed, and so was Warwick (Horton), my agent, really there was only one club in my mind.
"I'm just delighted to be settled now and I just wanted to concentrate on playing for Burnley.
"I've got to fly the nest and show people what I can do because it's no good playing one game every two months and playing the rest for the reserves.
"I need to be playing competitive football, and the English league is, I think, the best there is.
"The gaffer was really positive, and the faith he had in my led me to believe this is the right club for me."