The Hoops held off a late attempt by PSV Eindhoven to hijack the £3million deal they already had in place with Sporting Gijon for the Serbian striker.
Scepovic had been informed by the Spanish Second Division side he would be allowed to leave in this transfer window.
Peter Lawwell and his negotiating team have been working for the past four days to persuade the 24-year-old hitman that moving to Parkhead was his best option.
He has now accepted and has travelled to Paris to undergo a medical and wait for the work permit he needs to join up with Ronny Deila's squad.
The Celtic manager admitted the tall, powerful attacker ticked all the boxes for the man he wants to lead his line and provide the goal threat which was so sadly lacking in the Champions League Play-offs against NK Maribor.
The fee Celtic put on the table will be the first money Deila has spent on a new signing since he took over from Neil Lennon two months ago.
The Hoops did manage to finally cut through all the red tape and get Wakaso Mubarak in yesterday, as first revealed on the Evening Times website.
However, the Ghanaian winger is initially here on loan for a year, with Celtic holding an option - reported to be £4m - to buy him from his Russian club, Rubin Kazan.
The Wakaso deal mirrors that which saw Jo Inge Berget arrive on loan from Cardiff and Aleksander Tonev from Aston Villa.
Jason Denayer has also been brought in on a season-long loan from Manchester City, but there is no right-to-buy attached to the deal for the 19-year-old Belgian defender.
Deila has insisted: "It must feel right in my stomach," before he will commit to a purchase, and forecast that it would only be in the final few days of the transfer window that the big business would be done.
In terms of outlay, Scepovic represents that big business.
But, progress on the deal appeared to have stalled in the wake of Celtic's failure to get into the Champions League, and only a concerted effort in the past 24 hours to get the deal done has delivered the man Deila wants.
It would have been another major blow if Scepovic joined the list of players who got away.
Haitian midfielder, Jeff Louis rejected the chance to move to Parkhead after the Hoops had agreed a price with his French club, Nancy.
Louis chose to move to Belgian club, Standard Liege, despite the fact Celtic claim their offer was more lucrative.
The Hoops were also reported to have been in the hunt to sign Polish striker, Lukasz Teodorczyk, but he has now completed his £3.6m move from Lech Poznan to Dinamo Kiev.
NK Maribor's creative midfielder, Agim Ibraimi - a thorn in the side of the Hoops in both legs of their Champions League Play-off - has also indicated he rejected an approach from the Scottish champions before the clubs fought it out for a place in the Group Stage.
Missing out on Scepovic would be an even bigger blow to Deila than failing to get any of these others as the window closes on Monday night, with squads for the Europa League Group Stage scheduled to be submitted to Uefa by then.
If Celtic had been forced to move onto another target on their list at this late stage it would have left them exposed to auctions and inflated demands.