The only goal of the match came two minutes before the break when Danny Swanson guided the ball home with a smart left-foot finish.
It was United's fifth game in 12 days but they showed no signs of tiredness against the Highlanders as they won five consecutive league games for the first time in 14 years.
United made only one change, with Swanson replacing the injured Sean Dillon, while Scott Severin was listed among the United substitutes following his return after a five-month lay-off with an ankle injury.
The Caley Jags, who ran out 4-0 winners at Tannadice back in August, made two changes from the side that lost 2-1 to Celtic in the Co-operative Insurance Cup in midweek.
Jonny Hayes and Eric Odhiambo were both given starts in place of the absent Nick Ross and Shane Sutherland, who dropped to the bench.
Johnny Russell looked in confident mood following his double in United's 2-0 win over Hearts but he was unable to guide his header from a Keith Watson cross under the crossbar with less than four minutes gone.
The visitors should have opened the scoring after nine minutes when Hayes found Richie Foran unmarked at the back post but the Caley skipper's header was comfortably gathered by Dusan Pernis.
United carved out a decent opening after 15 minutes when Swanson rolled the ball into the path of David Robertson but the midfielder's effort from the edge of the area drifted harmlessly wide.
Slovakia stopper Pernis then saved bravely at the feet of Chris Hogg as both sides continued to press for the breakthrough.
A Swanson effort from the edge of the box was deflected wide before United had a penalty claim waved away after Scott Robertson collided with Foran in the box.
Ryan Esson came to Caley's rescue after 40 minutes when he pulled off a smart save to deny David Goodwillie after the United striker had glided past Hogg with ease.
But the inevitable opener came two minutes before the break when Scott Robertson nodded the ball into the path of Swanson, who guided the ball into the bottom corner with his left foot.
Ross Tokely picked up the first booking of the match after 50 minutes for a reckless tackle on Goodwillie before the hosts had another claim for a penalty turned down when Swanson looked to have been impeded by Hogg.
The visitors made their first change after 56 minutes with Alex MacDonald coming on for Adam Rooney.
United stopper Pernis was at full stretch to keep out a Lee Cox drive from the edge of the area before David Robertson missed a glorious opportunity at the other end when he somehow fired over from six yards out.
Russell should have put the game beyond doubt after 76 minutes but he fired the ball well over the crossbar when he had time and space to take an extra touch.
However the miss was not to prove decisive as Peter Houston's men hung on for all three points.