Barry Smith's struggling side fell 18 points behind County, who won thanks to Paul Lawson's 80th-minute strike which was followed by substitute Gary Glen's injury-time goal.
It now leaves them facing a huge task to avoid relegation while County continue to soar onwards and upwards in their first experience of the SPL.
This was a must-win game for the bottom team, who already trailed their visitors by 15 points in their desperate battle to beat the drop.
They were buoyed by last weekend's Scottish Cup win over Morton, which saw them set up a mouth-watering quarter-final with bitter rivals Dundee United.
But they could not afford a repeat of their last Friday night experience, which was one to forget as they slumped to a 4-1 defeat at home to Inverness back in October.
And they almost got off to another nightmare start inside the first minute of this clash with the Highlanders.
On-loan Celtic defender Lewis Toshney was short with an attempted back-pass to goalkeeper Rab Douglas.
County midfielder Rocco Quinn was quick to latch on to the opening but just as he looked set make the hosts pay, Douglas raced off his line to clear the danger with a thumping clearance.
The visitors went into this game unbeaten in 2013 so were not lacking any confidence.
That was shown on 13 minutes when midfielder Iain Vigurs left Dundee defender Gary Irvine for dead wide on the right.
Vigurs then cut inside and steadied himself before unleashing a left-foot shot which Douglas was relieved to see clear his near post.
That was enough to sting the hosts into action and minutes later they mounted their first real attack.
Brian Easton's clipped pass allowed Colin Nish to run clear into space down the left.
The former Hibs and Kilmarnock man found himself in a decent position but his final ball into the box could not pick out any of his team-mates at the far post.
The visitors remained slightly on top and midway through the first-half they threatened the Dundee goal again.
This time Nish's attempted chested lay-off caused a move to break down and Alex Cooper wasted no time in seizing possession.
The former Liverpool player drove forward and rifled in a low 25-yard drive which flashed narrowly past Douglas' left-hand post.
Dundee had Douglas to thank for keeping them on level terms just after the half hour.
Richard Brittain's curling cross into the box was only half-cleared by the home defence.
The loose ball fell to County defender Grant Munro whose stinging half-volley was well saved by Douglas at full-stretch before Irvine swept the rebound clear at the expense of a corner.
But after 36 minutes they were claiming a penalty when Jim McAlister's shot appeared to strike Scott Boyd's hand at point-blank range, however referee John McKendrick was close by and waved play on.
There was little incident in the second half until Lawson struck with a fierce strike which Douglas could not keep out and Glen made sure with a cool finish to round the Dundee keeper with the last kick of the game.
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- Paul Lawson