The SPL side made the best possible start by taking the lead in the 10th minute. Scott McDonald burst down the left and his cross reached Darren Smith who saw his shot blocked. The ball fell kindly for Brian Kerr to rifle high past Paul Mathers from 12 yards.
Morton took a while to respond but they almost equalised on 28 minutes when a swerving shot from Scott McLaughlin from 22 yards was beaten out by Graeme Smith.
Well doubled their lead on 33 minutes when McDonald cut in off the left wing, held off a couple of defenders and fired a low shot past Mathers.
The second half was a poor affair with Motherwell looking to play within themselves and do just enough to make it through, while Morton struggled to make an impact on a steady home defence.