In a game contested by two very young sides at Almondvale, schoolboy Ross Smith scored the only goal of the game to give the Tangerines their second win on the road inside 48 hours.
Generally, it was a game of few chances, but United's winner midway through the first half followed a flurry of corners.
Indeed, Smith almost scored from the first one. Gordon Pope then had a header cleared off the line and from the resulting third corner Smith headed high into the roof of the net from 12-yards.
Fittingly, Sean Fleming supplied the cross, because like Smith, he had too had an outstanding game.
Overall, it was a very good team effort from the visitors against an equally young Motherwell side, who would have snatched a draw right at the death, but for an outstanding one-handed save by United keeper John Gibson.
Dundee United: Gibson, W Russell, Fleming, R Smith, Pope, Ryan McCord, Allan, Anderson, Mathieson, Ross McCord (K Smith 77), Dow (Lee 89).
Subs not used: J Russell.