County swept into the attack and David Winters crossed for Don Cowie but his shot was blocked.
Dundee had their first effort on goal a couple of minutes later when Tom McManus drove the ball from outside the area but Stuart Garden gathered easily.
But it was the home side who went ahead in the eighth minute when a deep ball from Fergus Tiernan was chested down by Alex Burke and Rankin drilled it past 16-year-old debutant goalkeeper Scott Murray for the opener.
County were controlling the middle of the park, but Garry Brady tried a speculative shot from outside the 18-yard box but it finished up well over the crossbar.
Winters had a chance to add to the lead on half hour but he failed to get enough power on his header following a Rankin corner.
At the other end, Steven Craig chased a long ball to the edge of the area but Stuart Garden was quick off his line to clear the danger.
But County were in the ascendancy and a Cowie corner was deflected on to the crossbar by Stephen McNally and behind for another corner, then Lionel Djebi-Zadig glanced a header past from Rankin's flag-kick.
The pressure paid off and County went 2-0 up in the 41st minute when Rankin and Burke linked up on the edge of the box and a neat back flick from Burke set Rankin clear to slip the ball past Murray.
Dundee full-back Paul Dixon limped off just before the break and was replaced by Iain Anderson.
Dundee made a positive start to the start of the second half and Simon Lynch made space for himself but he dragged his shot wide.
Then Steven Robb blasted over as the visitors tried desperately hard to get back into the game.
Winters had another chance to score when he got on the end of a Burke delivery but he headed wide.
But the next goal was not long in coming as another fine cross from Cowie was headed in at the far post by Rankin to complete his hat-trick.
Dundee were finding it difficult to get out of their own half but substitute Andy Ferguson tested Garden with a low grounder which the County keeper did well to turn round the post for a corner.
McManus had a half chance in the box but his goalbound shot was charged down by Martin Canning.
Steve McGarry replaced Rankin in the final minute as County ran out comfortable winners.