The in-demand 23-year-old midfielder was also a target for Rangers – a club he grew up supporting – but Celtic had their bid accepted late on Thursday evening before the player’s medical on Friday ahead of the move.
Rangers had, in fact, failed with three bids for Allan, but the former West Brom player will now turn out for Ronny Deila’s side in the Ladbrokes Premiership.
He told the Celtic website: “I am absolutely delighted to sign for Celtic, such a fantastic club.
“It is such an honour to be given this opportunity and I can’t wait to get started and play my part in bringing success to Celtic and our supporters.”
Celtic offered £275,000 plus former Easter Road loanee Dylan McGeouch on a permanent deal as well as another midfielder, Liam Henderson, on loan.
Earlier this summer Allan put in a transfer request in an effort to force through a move to Ibrox, but Hibs remained resolute throughout the saga, insisting they would not sell to a rival Championship club.
Rangers are their biggest opponents in the title race but with Celtic in the Premiership, they had no issue with letting Allan go there.