The 22-year-old recently had a trial spell at Turf Moor and impressed enough to earn a contract.
No fee has been disclosed but it is believed that the Irish club were hoping to gain around £200,000 from letting Lafferty go.
The Northern Irishman has had a previous spell at Celtic and it was this pedigree that pleased Clarets boss Eddie Howe.
"He's had a good grounding at Celtic. You can see he's had a good education. He's got a lovely left foot and he loves to get forward," said Howe.
"He's a really good footballer and he fit very much into our philosophy. He's one for the future but is also in our thoughts for now."
Lafferty has made five appearances for the Northern Irish under-21s.