SPL Falkirk came desperately close to becoming the latest victims in a year of amazing Cup shocks, but with 11 minutes left and first division County looking comfortable, Australian striker Daniel McBreen scored at the second attempt after his first shot had been parried.
The first chance of the game fell to Falkirk when Pedro Moutinho latched on to a clever pass from Steve Thomson only for goalkeeper Stuart Garden to make an important save with his legs.
Moutinho was soon set up again by McBreen, but he failed to hit the target from a tricky angle.
The first sign of concern for Falkirk came after 15 minutes when County striker Sean Higgins fired over. Two minutes later, however, Don Cowie came forward from midfield and left the home defence standing as he slipped the ball under the keeper Alan Ferguson from 12 yards.
Falkirk hit back and defender Kenny Milne went closest to grabbing an equaliser with a fierce long-range effort.
McBreen ensured a dramatic start to the second half with a left-foot shot from the edge of the penalty box that hit the top of the bar.
County also struck the woodwork four minutes later when top scorer John Rankin rattled the post with a spectacular effort.
The arrival of substitute Alan Gow for defender Craig Ireland gave the home side three men in attack for the last 17 minutes. It finally paid off for them.