Morton looked destined to leap to sixth in the table with a first away win of the season before the County defender struck four minutes after the 90 minutes.
First half goals from Brian Graham and Allan Jenkins looked like being enough to give the Greenock side a much-needed victory after Andy Barrowman's opener.
But Boyd struck in time added on with a header from Michael Gardyne's corner, much to the outrage of the visitors who protested at the amount of stoppage time.
McStay, having waited three weeks for his first match through freezing weather, found two games coming in quick succession after Saturday's 1-1 draw at Partick.
The ex-Celtic player and coach's first game at Victoria Park was to offer much food for thought, though.
County's confidence looked fragile at times and the hosts were prone to errors which doubtless added to the confidence running though the Morton side.
Just as it looked to be over for the home side, though, Boyd mustered the second draw of the McStay era and keep County seventh in the table.