Jason Walker's 49th minute header was enough to beat Kettering on Saturday and send Barrow into Round 2 of The FA Trophy.
Tim Deasy made three fine saves after being called into the side in place of Stuart Tomlinson, who has sore ribs.
Kettering dominated early on, but Barrow held territorial control after that with Robin Hulbert and Carlos Logan going close before half-time.
Marc Goodfellow's inch perfect cross picked out the unmarked Walker for him to head home, while later Goodfellow brought a fine save from Cooper and Jordan Bowery had a shot cleared off the line while Simon Spender shot over from the rebound.