Robbie Neilson's men made it four wins in a row to move six points clear of closest challengers Ayr United and nine in front of their city rivals.
The first half produced little in the way of goalmouth incident but Dundee United went ahead in 56 minutes when Nicky Clark scored from the spot.
SPFL top scorer Lawrence Shankland netted his 17th Ladbrokes Championship goal of the season to seal the win for the visitors.
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