Harker grabbed the only goal 15 minutes from the end when he powered home an inch-perfect cross from Bobby Thomas.
It was the striker’s second goal of the pre-season programme after he had opened his account to help beat Polish side Pogon Szczecin II at the end of the Clarets’ training camp.
And it was enough for Steve Stone’s side to make amends for defeat to BetVictor Premier League opposition a week ago.
The Clarets failed to compete with a more robust Stalybridge Celtic side as they lost 3-1 at Bower Fold.
But this time they showed the necessary resolve to get past Celtic’s league rivals as Adam Legzdins kept a clean sheet on his return to Under-23 duty.
Helping lend some useful experience as an over-age player, the goalkeeper had little to do against a Bamber Bridge side which just avoided the drop last season.
But he did provide a calming presence to a much-improved defensive display and made one tidy save just before half-time when gathering a shot from Brig midfielder Danny Forbes.
The Clarets didn’t create many chances themselves, an early opening for trialist striker coming to nothing as Lloyd Rigby dealt with a tame finish.
The hard-working Harker did have the ball in the net before the break, but his cross-shot was chalked off for off-side.
And it was Thomas who went closest to the opening goal when his header from a Joel Senior cross was saved at the foot of a post by Rigby after almost half an hour.
The Clarets, who included a handful of trialists, had plenty of possession after the break in a competitive affair.
But just as though it looked as though they may have to settle for a stalemate, Thomas popped up on the right flank to deliver a perfect cross for Harker to bury a towering far-post header back across substitute keeper Danny Roccia and into the far corner to prove the difference.
Bamber Bridge: Rigby (Roccia HT); Churchman, Doughty (Ross 66), Charnock, Sumner; Booth (White 63), Forbes (Marlow 77), Wisdom (Spooner HT), Pett (Buchan 80); Harries (Fofana 80), Waddecarr (Dudley HT).
Burnley: Legzdins; Cropper, Thomas, Younger, Wilson; Senior (Trialist 52), N’Guessan (Trialist 61), Trialist (Steels 68), Conn-Clarke (Trialist 66); Harker (Richardson 77), Trialist (Harris 56).
Referee: Lewis Smith