Oli Shaw and Billy McKay both netted for County inside two first-half minutes and Montrose hit the frame of the goal twice before Regan Charles-Cook netted an impressive third for the visitors six minutes into the second half.
But Montrose turned the game on its head with two goals in four minutes from Graham Webster and on-loan St Johnstone midfielder Cammy Ballantyne and Andrew Steeves netted from eight yards with a minute left.
County gained a second point with a 5-4 sudden-death shoot-out win following the 3-3 draw but manager Stuart Kettlewell will not have been appeased over their sloppy second half.
Shaw opened the scoring in the 13th minute when his deft touch on the edge of the six-yard box sent McKay's effort beyond the goalkeeper.
McKay soon stooped to head home Charles-Cook's ball across the face of goal before County had two let-offs.
Former Dundee United skipper Sean Dillon hit the bar from a free-kick five yards inside his own half after spotting Ross Laidlaw off his line, and the goalkeeper then fumbled Ballantyne's 25-yard strike against his post.
Craig Johnston twice came close for Montrose after dangerous balls into the box before the break.
The visitors seemed to have settled matters when Charles-Cook collected a loose ball off his own throw-in, drove inside and curled a left-footed effort inside the far post from 20 yards.
However, Webster rose highest to head home Steeves' cross in the 63rd minute and McKay squandered a great chance when he lobbed against the bar and saw his headed rebound cleared off the line.
Ballantyne's harmless effort was diverted into his own goal by Callum Morris, who soon headed off the Montrose post.
The hosts levelled when left-back Steeves netted after a goalmouth scramble to earn Montrose a point but they could not double their return in the shoot-out.