Chris Humphrey came close to making it 2-0 to Motherwell just minutes later with a powerful drive from inside the box, but Cierzniak made a good save.
However, the United goalkeeper failed to do anything in the eighth minute when Henrik Ojamaa scored his side's second goal of the evening, turning away from Sean Dillon before rolling the ball into the net.
Randolph pulled off another important save before the break, this time keeping out Gary Mackay-Steven's deflected effort from inside his area.
The visitors went 3-0 up 10 minutes into the second half, Higdon scoring his second goal of the match by getting the better of Dillon and then shooting past Cierzniak.
Minutes later United were denied by the frame of the Motherwell goal when Willo Flood's free kick from 25 yards clipped the outside of the post.
Jon Daly pulled one back for the home team on 69 minutes, but it was too little too late.