Ryan Dow opened the scoring at Tannadice, before Gunning - who has been sidelined in recent weeks by a knee injury - netted either side of the interval.
John Sutton pulled a goal back from the penalty spot for the visitors, but United saw the game out to close the gap between themselves and third-place Well to six points.
Jackie McNamara's side fully deserving of a third successive victory, although Motherwell were way below the standards they had set themselves in recent weeks during a run of seven wins in nine games.
From the moment Dow opened the scoring in the 22nd minute, when he collected a misplaced back-pass from Shaun Hutchison and swept a shot past Gunnar Nielsen, United seized a control on the game they refused to relinquish.
However, it was not until three minutes prior to the break that they added to their lead as Gunning stabbed home following a Nadir Ciftci flick-on from Dow's corner.
Motherwell could have no complaints over the scoreline but they were aggrieved at being denied the opportunity to narrow it in the 44th minute when Gunning appeared to go through Iain Vigurs in the box only for referee Bobby Madden to wave away calls for a penalty.
Things were to get worse for the visitors at the start of the second half when they found themselves trailing further as Gunning's 25-yard free-kick, awarded after Vigurs had brought down Ciftci, found the net via a deflection off the wall.
There appeared no way back for Motherwell, but they were given a glimmer of hope when Ciftci was penalised for handball in the box and Sutton converted the resulting penalty.
However, the hosts closed the door on their visitors and played out the rest of the game in relative comfort.