McDermott then teed up Daryl Clare from close range, but Andy Bond raced on to Goodfellow's pass to slot home.
Clare poked home his second from Craig Baxter's assist while Gregg Blundell tucked away Barrow's third equaliser.
Barrow: Tomlinson, Spender, Jelleyman, Bolland, Pearson, Goodfellow, Hulbert, Bond, Walker, Blundell (Bowery 76), Nelthorpe (Rothery 88). Subs Not Used: Deasy, Jones, Boyd.
Booked: Spender, Bolland.
Goals: Goodfellow 4, Bond 51, Blundell 60.
Gateshead: Farman, Baxter, Robinson, Gate, Curtis, Phillips, Swailes, McDermott, Williams, Clare (Richardson 83), Shaw. Subs Not Used: Provett, Turnbull, Armstrong, Cave.
Booked: Gate.
Goals: McDermott 2, Clare 42, 57.
Att: 1,727
Ref: R Metcalfe (Lancashire).
Source: BBC Sport
Source: BBC Sport