Northern Ireland international Lafferty, 20, has signed a five-year contract with the Scottish Cup and CIS Insurance Cup holders.
He told rangers.co.uk: "I'm delighted to be here. The move has been a long time in coming and I'm thrilled to be at the club at last.
"When I heard Rangers were looking at me, I really wanted to join them but there was a point when I thought they had lost interest.
"I heard nothing from them so I was worried the move wouldn't happen and I was delighted when my agent rang me to say Burnley had accepted the bid for me.
"Hopefully I can now show the fans what I can do on the pitch. All of my mates and family are big Rangers fans so they're pleased I'm at the club now."
Lafferty only scored five league goals for Burnley last season but he is considered one of Northern Ireland's hottest prospects in years and will challenge the likes of Kris Boyd, Jean-Claude Darcheville and Nacho Novo, plus his two fellow new boys, for a place in Walter Smith's Rangers attack next season.
He admits it was the night of the UEFA Cup final which convinced him he should sign for Rangers, if the opportunity came up.
"I was at the final in Manchester and the atmosphere there was incredible," he said.
"That helped me make up my mind that I wanted to play for Rangers. The fans were amazing then and playing in Europe is a big attraction for me.
"I want to play at that level and I'm being offered that here."