Woods then crossed for Martin Hardie to grab Dunfermline's second just before the half-time interval.
Kallum Higginbotham was brought down in the box by Woods and Mark Millar converted the penalty on 58 minutes but the visitors held on for the win.
Falkirk: Olejnik, Perry, McLean, Scobbie, Twaddle, Higginbotham (Moutinho 76), Murdoch (Compton 46), O'Brien, Millar, Stewart, Finnigan (McManus 46). Subs Not Used: Barclay, Wallace.
Booked: Millar, Stewart.
Goals: Millar 58 pen.
Dunfermline: Chris Smith, Woods, Higgins, Keddie, McCann, Hardie (Thomson 63), Burke (Bayne 76), Mason, McDougall (David Graham 77), Cardle, Kirk. Subs Not Used: Paterson, Buchanan.
Booked: Woods, McCann.
Goals: Woods 28, Hardie 44.
Att: 5,121
Ref: E Smith
Source: BBC Sport
Source: BBC Sport