After a goalless opening 45, Clarke struck a rocket into the top corner after pressuring Bromley defender Chris Bush.
The visitors equalised when Jude Arthurs made goalkeeper Grant Smith pay for a rare mistake.
But it did not matter in the end as Carline finished at the back post following good work by Joel Taylor.
Here are Liam Norcliffe’s player ratings from the game...
1. Grant Smith 6
A rollercoaster of a night for the stopper. Fresh from signing a new one-year contract, Smith's rare error gifted Bromley an equaliser but he redeemed himself with a superb save low down to his left late on. It was a top stop. Photo: Dean Atkins
2. George Carline 9
A man of the match contender. Almost grabbed an assist in the first-half when Taylor volleyed his cross against the woodwork. Taylor and Carline combined again three minutes after half-time but the latter's effort was cleared off the line. But he wouldn't be denied next time when he finished expertly at the back post when Taylor picked him out again for his first Chesterfield goal. After the game boss Rowe said Carline runs 11.5km a match. Wow. Photo: jason chadwick
3. Fraser Kerr 8
A no-nonsense display from the centre-back who hardly put a foot wrong. Filled in well for Gunning when he hobbled off with 20 minutes to go. Photo: Craig Brough/AHPIX LTD
4. Gavin Gunning 8
A commanding performance at the heart of the Town defence. He is clearly not 100% fit but he is putting his body on the line for the team. Kept top scorer Cheek quiet. Went off with a tight hamstring with 20 minutes to go. Photo: Craig Brough/AHPIX LTD