Hampshire's low spot-kick following a soft hand-ball decsision against Craig Brittain was well saved by Stephen Grindlay before being cleared by the Sons' defence.
Brechin had plenty of chances to build a healthy lead during a wind-assisted first half, but Paul Ritchie was their major sinner as they failed to make their pressure count.
The Sons looked more likely after the break, and it was no surprise when they took the lead from Iain Russell's crossfield pass which picked out substitute John Dillon, and he fired under Craig Nelson from the edge of the box.
City got themselves level against the run of play when James Sharp stole in front of the home defence to hook home a Charlie King corner.
A late push by Brechin saw substitute Graham Gibson sent clear, but he was denied by Grindlay while Hampshire drove an effort off the inside of the post when it seemed easier to score.