These two teams finally got the tie underway at the fifth attempt despite the pitch having a covering of snow.
The referee allowed the game to go ahead after the lines had been cleared and coloured orange.
His decision may have been influenced by the presence of Scotland's First Minister Alex Salmond, who happens to be the local MSP.
It was the visitors who got off to the perfect start scoring after only 90 seconds.
Jim O'Brien floated a free-kick into the six-yard box where John Sutton rose unchallenged to head past Andy Reid.
It was one-way traffic throughout the first half and the major surprise was that Well did not add to their lead.
It had become a personal duel between Reid and Sutton with the keeper pulling off a number of fine saves to deny the Englishman.
The home side started the second half pushing forward and Neil McLean was unlucky to see an 18-yard drive slide wide of the post.
Motherwell then went up the park and made it two on 57 minutes when David Clarkson headed in at the near post from an O'Brien cross.
Halfway through the second half the SPL side killed the tie off when Sutton tapped home Stephen Craigan's cross from only a yard out.
In the end, Motherwell avoided any potential slip-up with a professional performance and progressed to take on St Mirren at home in the fifth round.