The Staggies, now unbeaten in 38 league games, netted the winner with less than 20 minutes remaining after Mark Corcoran had been impeded by Neil McGregor.
The defeat leaves Dundee at the foot of the table with only one point from their opening four matches.
Former Hibernian striker Colin Nish, who has joined Dundee on loan from Hartlepool, was handed his debut as the Dark Blues made three changes from the side defeated 3-0 in last weekend's derby against Dundee United.
Defender McGregor and former Falkirk attacker Mark Stewart were also brought in with Matt Lockwood, John Baird and the injured Nicky Riley making way.
There was also a place on the bench for Scotland Under-21 defender Lewis Toshney, who has joined on a season-long loan from Celtic, while County stuck with the same side that was only seconds away from beating the Hoops last weekend.
The hosts had a lucky escape after six minutes when Ross Tokely nodded an Iain Vigurs corner on to Scott Boyd, whose flick-on bounced back off the crossbar.
Dundee skipper Stephen O'Donnell, who had successfully appealed during the week against his sending-off against Dundee United, was unfortunate to see his well-struck left-foot volley blocked by Vigurs for a fruitless corner.
The Dark Blues were in the ascendancy when Kyle Benedictus surged down the wing, leaving a trail of County defenders in his wake, before picking out Nish in the centre but the striker's header failed to trouble Michael Fraser.
The Dundee pressure continued when Ryan Conroy played a neat ball into the path of Nish, who could only find the side-netting with his first-time effort from an acute angle, before Stewart headed wide from a Conroy free-kick.
And Barry Smith's men went even closer on the stroke of half time when Gary Irvine out-muscled two County defenders to get on the end of a Jim McAlister cross before thundering a left-foot drive off the upright.
Toshney made his first appearance in a Dundee shirt three minutes after the break when he replaced the injured Benedictus.
Colin McMenamin, playing against his former side, almost created the opening for County when he threaded an intelligent ball into the path of onrushing midfielder Stuart Kettlewell, but his cut-back was too far in front of Vigurs and the chance was gone.
Brittain, who netted an exquisite set-piece against Celtic last weekend, drew a fine save from Robert Douglas from a free-kick 30 yards out as the visitors kept up their efforts to make the breakthrough.
However, it was to be the County skipper who opened the scoring in the 72nd minute when Corcoran was brought down in the area by McGregor and Brittain clinically fired the ball home from the penalty spot.
Dundee pushed for an equaliser late on but were unable to get any joy against the well-organised County defence as the Highland club held out for maximum points.