But Derek Lyle levelled for Morton just before the hour mark with an impressive strike from the edge of the box.
And Allan Jenkins completed the comeback with eight minutes remaining, driving low into the net.
Visiting keeper Michael McGovern pulled off a finger-tip save to deny Stewart Kean after seven minutes.
That was the best chance of the game until Craig smashed home Mark Corcoran's cross to give County a 1-0 lead at the interval.
But it was the home side who came out fighting after the break and, for the second week in a row, Lyle scored a wonderful goal with a strike into the top corner to pull the hosts level.
That was just before the hour mark and, as the game swung from end to end, Jenkins struck past McGovern from 18 yard for the winner.
Morton: Stewart, MacGregor, Smyth, McCaffrey, McKinlay, Kean, Jenkins, Tidser, O'Brien (Monti 60), Lyle (Holmes 87), Graham (Weatherson 57). Subs Not Used: Cuthbert, Bachirou.
Goals: Lyle 59, Jenkins 82.
Ross County: McGovern, Marr, Boyd, Gartland, Morrison, Brittain, Kettlewell (Smith 87), Fitzpatrick, Corcoran, Wood, Craig (Di Giacomo 46). Subs Not Used: Malin, Flynn, Vigurs.
Booked: Kettlewell, Gartland, Brittain.
Goals: Craig 38.
Ref: C Richmond
Source: BBC Sport