The Russian forward produced a brilliant strike from outside the box as he collected Gary Mackay-Steven’s assist to find the top left-hand corner at Tannadice.
United just about deserved the three points although the visitors did have their moments.
With just three minutes on the clock, Motherwell’s Josh Law struck the bar from 22 yards before the influential Mackey-Steven missed a great opportunity after creating an opening for himself.
Fraser Kerr headed over from close range while, at the other end, John Rankin was off target when well placed.
Mackay-Steven continued to impress as United finished the game stronger.
Bilate headed wide from close range before the same player produced the decisive moment on 82 with a superb effort from 25 yards.
In the third minute of injury time Craig Reid’s attempt to force an equaliser was blocked as United held on.