Brown put the home side ahead with a 20-yard effort.
And Clancy doubled the score when he fired into the roof of the net from 12 yards to leave Forest Green just two points above the relegation zone.
Fleetwood Town: Davies, Connors (Seddon 46), McNulty, Hining, Brown, Barry, McGuire, Milligan, Cavanagh, Vieira (Rowe 75), Donnelly (Clancy 46). Subs Not Used: Hurst, Thorpe.
Booked: Cavanagh, McGuire.
Goals: Brown 56, Clancy 80.
Forest Green: Bittner, Hodgkiss, Luke Jones, Hall, Armstrong, Dyer (Gray 81), Caines (Guinan 57), Turk, Forbes, Klukowski (Enver-Marum 69), Styche. Subs Not Used: Bartlett, Allen.
Booked: Guinan.
Att: 1,335.
Ref: B Khatib (England).
Source: BBC Sport
Source: BBC Sport