Dods powerfully headed home a Barry Robson corner in the 78th minute to settle a tight match that could have gone either way.
However, Craig Levein's side did just enough to remain unbeaten at home this season and they are now only three points adrift of Champions League heroes Rangers in third place.
Levein made two changes to the side that lost against Celtic last weekend with Christian Kalvenes and Steven Robb drafted into the starting line-up in place of David Robertson and Jordan Robertson.
And it was the home side that made the early running with Robson and Prince Bauben going close before Noel Hunt saw a close-range chance snuffed out by the agility of defender Jim Paterson.
Hunt then tried his luck again in a first period devoid of much goalmouth action with a shot that was expertly turned away by Graeme Smith.
The second half brought an improvement from the visitors and Stephen Hughes saw a close-range shot blocked by Morgaro Gomis.
The Steelmen were on top now and Ross McCormack was inches away with a curling effort from the edge of the penalty area.
However, the home side responded in devastating fashion as they took the lead against the run of play.
With 12 minutes remaining Dods got up well to head home the only goal of the game before United closed out the match with few alarms to add another three-point haul to their impressive season's total.