Pressley isn't the type to get carried away and he stressed before kick-off at Tannadice that his team can still improve in multiple areas, but this was a game they would not have won earlier in the campaign.
They stood firm in the second half, putting bodies on the line as United turned the screw.
Koumetio was superb at the back, making 15 clearances and winning nine of his 12 aerial duels and he was backed up by centre-back partner Luke Graham.
When the Dundee defence was breached, McCracken was there to bail them out, with the save from Stephenson simply outstanding.
Hay led the line well in addition to his match-winning goal and was supported by Yan Dhanda, who delivered an excellent all-round display, putting dangerous balls into the box and doing the dirty work going the other way.
Not only have they won three in a row, they are now unbeaten in their past four trips to Tannadice and have moved nine points above Kilmarnock in the relegation play-off spot.
United boss Jim Goodwin will look back with regret on the Camara error that led to the goal and the chances they wasted in the second half.
From 42 touches in the Dundee box, they only managed five shots on target - a poor conversion rate by anyone's standards.
One moment late on summed up their day. Amar Fatah wriggled past a couple of defenders before teeing up the usually reliable Zac Sapsford, but he took a needless touch and was unable to get his shot away before being swarmed by dark blue shirts.
Performances like that will deliver points more often than not, but it is results, rather than performances, that matter on derby day.