Results elsewhere in the Ladbrokes Premiership - with Partick Thistle losing to Motherwell - mean the Staggies remain two points away from a play-off spot and cannot escape the bottom two.
Simon Murray grabbed the decisive goal at the Global Energy Stadium with just his third for Dundee and first since February, which means the visitors have confirmed their status in the top flight beyond all doubt.
The defeat means County must beat St Johnstone on Saturday and hope Thistle fail to win if they are to avoid automatic relegation.
The hosts flew out of the traps and were almost ahead after just 90 seconds, only for Billy McKay to be flagged for offside after stabbing in a free-kick at the far post.
The Staggies stayed on top for the opening 15 minutes but missed the best chance of the half when Davis Keillor-Dunn’s superb corner found Liam Fontaine at the back post. The defender ran on to the ball but could only volley wide before blaming the sun for getting in his eyes.
Dundee slowly grew into the game and ended the first half on top, but it was a game low on quality and severely affected by a series of full-blooded challenges.
The Staggies came flying out of the blocks again after the break and should have scored in the first minute of the half.
Jason Naismith did well to lift a ball in to the far post from the byline to an unmarked Keillor-Dunn but the forward tried to volley in when he could have stooped to head it and the ball sailed well over the bar.
They were made to pay and it was Randy Wolters - on his first league start since October - who created the goal.
The winger delivered a wicked cross from the left that bounced in the six-yard box and Murray reacted quickest to flick the ball in with his head.
Keillor-Dunn headed straight at Elliot Parish after drifting off his marker as the hosts threw men forward but they were struggling to make regular inroads as Dundee forced them out wide.
Their best chance fell to the experienced David Ngog who sliced wide and Michael Gardyne was booked for diving in the box.
Paul McGowan curled one against the bar at the other end and Murray missed a golden chance to wrap it up when one-on-one but the visitors held on to leave Ross County deep in trouble.