With a game left to play, United now has one point more than the Highland team than the Staggies and is in 11th place overall.
Liam Smith delivered a cross in from the right for United in the fifth minute, and Ryan Edwards rose high to head it just wide of County goalkeeper Ross Laidlaw's far post.
Four minutes later, though, the home team had the lead. IGlenn Middleton on the left received a beautiful crossfield pass from Ian Harkes, and his low cross was deflected by Keith Watson before being turned in on the line by unfortunate teammate Randall.
Edwards then missed a fantastic opportunity for County to score an early equalizer when he raced through on goal and passed United goalkeeper Mark Birighitti with a low drive that went wide of the post.
When Mulgrew delivered a precisely placed free kick from the left, Steven Fletcher sprang high, but his glancing header struck the post and United came agonizingly close to doubling their lead.
Shortly after, a Watson header from a Dhanda corner that blasted off the post caused the Staggies to also strike the woodwork.
Just after the break, the Tangerines had two quick opportunities. First, Fletcher's fierce volley was blocked by Randall, and then the veteran striker saw another shot expertly saved by Laidlaw.
But in the 70th minute, United doubled their advantage. Laidlaw did well to deflect a Smith shot wide, but Mulgrew powered a header beyond the County goalie after Dylan Levitt delivered a wonderful corner kick.
The visitors believed they had equalized through replacement William Akio, but a VAR review determined the goal to be offside.
In the 85th minute, County was down to 10 men after referee Chris Graham issued Edwards a second caution. In the sixth minute of added time, Sibbald's curling strike from the edge of the box into the goal placed the cherry on top for the home team.
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