The Japan striker was on target at the end of the first half after the champions had taken the lead at Parkhead through an own goal from Ayo Obileye.
But first-half stoppage time saw Livingston captain Nicky Devlin pull a goal back in his 100th Premiership appearance.
Celtic dominated the first half but were less fluent after the interval, although Liel Abada, who set up both their goals, had a goal disallowed after a lengthy VAR review.
The Glasgow giants went ahead in the 23rd minute when Abada's powerful centre deflected off Obileye's knee and flew into the roof of the net from six yards out.
Furuhashi had two shots saved before he doubled the lead, the forward getting ahead of Obileye to turn in a low cross from Israeli star Abada for his 12th goal of the season.
Livingston, however, stunned the home crowd when, barely 90 seconds later, Devlin, capitalising on hesitancy in the Celtic defence, poked the ball past former England goalkeeper Joe Hart.
Former Rangers midfielder Stephen Kelly nearly equalised with a well-struck volley from 20 yards, his shot producing an acrobatic save from Hart.
Celtic thought they had a third goal when Abada fired home but the winger's effort was eventually disallowed for a push on Livingston defender Morgan Boyes.
Both Abada and Sead Haksabanovic then missed half-chances, but there were no late scares for Celtic as they extended their lead over arch-rivals Rangers at the top of the table.
