The former Chelsea striker headed a trademark winner after 51 minutes, and this result coupled with Chester's home defeat by Macclesfield means the Bees are safe for another year.
It was cruel on the Cherries who dominated for long periods, Lee Bradbury heading against the crossbar with Bees keeper Jake Cole stranded after 11 minutes.
Gary Breen then let in Brett Pitman with a dreadful back-header, but Cole bailed out his centre-back with a superb diving stop, and from the resulting corner the hosts survived again when Jason Pearce's header was cleared off the line by Neal Bishop.
Cole denied Pitman again early in the second half, palming a fierce free-kick behind, and it was against the run of play when Furlong headed Barnet in front.
Bournemouth refused to buckle, and it needed Joe Devera's clearance off the line on the hour from Jeff Goulding's scuffed effort to preserve the lead, before Cole kept out Bradbury late on to seal the win.