James Berry scored his first goal in two months when he blasted in off the post on 62 minutes after Lee Bonis forced a mistake from home goalkeeper Wyll Stanway and that turned out to be the winner in Cumbria.
The results means the Spireites, who stay seventh in League Two, have won all of their last three matches and kept clean sheets in all of them.
Here are our player ratings from the game...
5. Malik Owolabi-Belewu 6
Malik Owolabi-Belewu 6 Reasonably steady apart from one loose misjudged header which went behind for a corner. Photo: Tina Jenner
7. Sammy Braybrooke 6
Sammy Braybrooke 6 If anyone was going to open up the low block of Barrow it looked like it would be him in the first 45. Made some important tackles. Photo: Tina Jenner
8. Liam Mandeville 6
Liam Mandeville 6 Played further forward this time out, firing just off target in the opening minutes. Disciplined display. Off for Markanday with 20 to go. Photo: Tina Jenner