Manuel Pascali gave Kilmarnock the lead against the run of play but Russell rewarded United’s first-half dominance with an excellent finish two minutes before the break.
The goal originated from Keith Watson and Russell headed United in front from the same source before completing his treble two minutes from time to move his side above Kilmarnock into seventh place in the Scottish Premier League.
Pascali headed a second goal coming in the final seconds, but there was no time to push for an equaliser.
The visitors made the better start and thought they had taken a sixth-minute lead when Russell headed home an excellent Barry Douglas free-kick at the back post but the flag was raised for offside.
Killie were struggling to get forward in numbers but they took the lead in the 18th minute following James Dayton’s corner. Ryan O’Leary volleyed towards goal from 20 yards and both Paul Heffernan and Pascali had chances to turn home from inside the six-yard box with the Italian making no mistake.
The pattern of play stayed the same with United dominating possession inside their opponents’ half. They continued to put the ball in the box with Jon Daly and Russell unable to head difficult chances on target and the former came closer with a turn and shot from 18 yards.
But the pressure paid off following another cross. O’Leary did well to hook Watson’s ball away from Daly but it fell for Russell who powered a first-time left-footed effort into the bottom corner from 16 yards.
United keeper Radoslaw Cierzniak parried Rory McKeown’s effort but the Tayside club went ahead in the 63rd minute.
The visitors worked the ball to Watson on the right wing and the full-back curled in a perfect cross to allow Russell to head down and into the net from six yards.
Shiels brought on attack-minded players Rory McKenzie, Borja Perez and Jude Winchester but United stretched the game well to keep the pressure off.
Russell sealed the points in the 88th minute when he ran on to a perfectly weighted ball over the top from Douglas before slotting the ball into the corner of the net.
Pascali headed home Dayton’s corner two minutes into stoppage time but there was barely time for United to kick off again.