Furuhashi also scored twice in last year's final and took his tally for the season to 24 with two close-range finishes either side of half-time.
Alfredo Morelos pulled a goal back 26 minutes from time but Rangers succumbed to a first defeat since Michael Beale replaced Giovanni van Bronckhorst as manager in November.
Victory could be the first of a domestic treble for Ange Postecoglou's men as they lead Rangers by nine points at the top of the Scottish Premiership and are into the quarter-finals of the Scottish Cup.
"He's a special player," said Celtic captain Callum McGregor of Furuhashi's contribution.
"It means the world to me. Time after time, you're tasked to win at this club."
Furuhashi missed his first couple of chance of the game as Allan McGregor rushed from his line to block before Celtic's talisman headed a glorious opportunity over.
However, he could not miss a minute before half-time as Daizen Maeda flicked on Greg Taylor's cross and Furuhashi was unmarked to tap in from point-blank range.
A fine team move set up his second as Reo Hatate burst into the Rangers box and squared for his compatriot to force the ball home 11 minutes into the second half.
At that point the only doubt over the outcome seemed to be Celtic's margin of victory.
But Rangers rallied after Morelos' strike had too much power for Joe Hart when Celtic failed to clear a free-kick.
Beale's men then surged forward in search of an equaliser to take the game to extra time.
But it was the Hoops who had the better chances to seal the trophy in the closing stages as McGregor denied Matt O'Riley and Sead Haksabanovic fired wide.
