The former Chelsea man marked a joyous but relieving day for the Bees, with a trademark header after 51 minutes.
His side are now mathematically safe after Chester's defeat, leaving Bournemouth still in the thick of relegation trouble.
It was the Cherries, though, who looked the more dangerous early on and should have found themselves two up within half an hour.
Liam Feeney sent a deep cross finding Lee Bradbury completely unmarked in the area. He beat the keeper but not the crossbar with a well placed header on ten minutes.
Moments later, a weak headed back-pass from Gary Breen let in striker Brett Pitman, who forced Jake Cole into a low save to his right, parrying the ball out for a corner.
From the resulting cross Bradbury rose unchallenged again, heading over the advancing keeper, only to have his effort booted off the line by Neal Bishop.
The away side controlled most of the first half with defenders Ryan Garry and Jason Pearce proving stronger than Furlong and his partner Adam Birchall.
Barnet looked inept and lacking in ideas, resorting to direct balls that were too easily dealt with by Bournemouth.
In the second half Barnet rallied, with captain Bishop the instigator. The central midfielder has been crucial for his team all year and epitomised his season with a headstrong second-half performance.
The much-deserved goal came as Furlong controlled and laid the ball off to Kenny Gillet, who whipped a vicious cross onto the head of Furlong, who nestled the ball into the far corner.
Bournemouth should have nicked a point five minutes before time as another weak back-pass put in Bradbury, whose shot was deflected over by the goalkeeper.