Most of the damage was done in the first half as United rattled in three goals without replay after a Gary MacKay-Steven penalty was followed up by strikes from Stuart Armstrong and Ryan Gauld.
Nadir Ciftci added a fourth mid-way through the second half before Ivan Sproule and Rocco Quinn scored consolation goals for County.
Ciftci was almost through in the second minute when Grant Munro misjudged a long through ball but Brian McLean was there to clear the danger.
Ciftci came close a minute later when United were awarded a free-kick in a dangerous position.
MacKay-Steven touched the ball to the Holland-born forward, whose shot was heading for the top corner before County keeper Mark Brown managed to tip it over for a corner.
It was United who had the upper hand and Gauld had a chance in the 15th minute when a Paul Paton free-kick fell for him in the box. But the youngster did not get enough power on his shot and Brown saved easily.
Two goals in the space of a minute turned the match United's way. The opener came via the penalty spot in the 18th minute after MacKay-Steven was taken out on the edge of the box by Brown.
MacKay-Steven took the penalty himself and sent Brown the wrong way from the spot.
The visitors doubled their lead 60 seconds later when Armstrong got on the end of Ciftci's chip into the box at the near post and his glancing header dipped over Brown and into the net.
County had a strong claim for a penalty in the 27th minute when Kevin Luckassen appeared to be hauled down in the box by Sean Dillon but the match official saw nothing amiss.
The home side were desperately unlucky not to pull one back in the 35th minute when a 25-yard free-kick from Graham Carey rebounded off the far post with Luckassen unable to steer home the rebound.
Seconds after Stuart Kettlewell fired wide of the near post United went up the other end and scored a third five minutes before half-time.
In a fast flowing move Gauld beat the off-side trap to fire into the far corner of the net after Ciftci and Stevenson combined in the build up.
County started the second half on the front foot but found it difficult to make the breakthrough.
In fact, it was United who almost increased their lead in the 58th minute when MacKay-Steven slipped the ball through to Ciftci but his shot was well saved by Brown.
County came close again when substitute Orhan Mustafi got on the end of a deep cross from Richard Brittain but his downward header was scooped past the post by United keeper Radoslaw Cierzniak.
But it was United who went further ahead in the 65th minute when Gauld threaded the ball through to Ciftci, who coolly tucked the ball away.
County never gave up the ghost and they pulled one back in the 71st minute when Sproule found the far corner of the net thanks to a wicked deflection.
It was game on five minutes later when Carey's pin-point cross from the left was headed powerfully into the net from six yards by Quinn.
But it was too little too late from the home side as United held out comfortably for all three points.
Dundee United manager Jackie McNamara was delighted with his side's display and purred about his youthful players.
He said: "Young Gauld at 17, his touch and awareness and ability is frightening.
"I see him every day in training and I know what he's capable of.
"The first question I was asked today was about David Goodwillie not starting and Ciftci and Gauld coming in, but they have answered that and shown that they are ready.
"I thought Nadir was crucial to how we played, he got the ball down under control to let the others support.
"There was a lot of pleasing things there."
It was not all smiles for McNamara, though, who was unhappy with the late goals conceded.
"They got a lift with getting a goal back from a deflection," he said.
"I felt for Rado (goalkeeper Radoslaw Cierzniak) because I thought he was excellent for us, with a few good saves in the first half.
"We showed a wee bit of inexperience at times, when you're four goals up you should keep the ball moving so we made it a bit nervy towards the end."
McNamara also believes there is a lot more to come from his young side, adding: "It's coming. I think you can see there that we could have taken more chances.
"The last pass or the decision making or the finish at times (needs work) but the pleasing thing was that they are getting better with the pace and the strength they've got."
County manager Derek Adams was unavailable for comment after the match.