The home stunned their Scottish First Division opponents as they raced into a 5-0 half-time advantage.
Tom Hateley netted directly from a corner before Jamie Murphy, Shaun Hutchinson, Henrik Ojamaa and Nicky Law all netted for the hosts as the tie was all but decided before the interval.
Murphy then blasted home his second midway through the second half to add to Morton's woes and complete a one-sided scoreline.
Motherwell had a glorious opportunity to take the lead after just four minutes when Morton goalkeeper Colin Stewart failed to deal with a back-pass. The ball broke to Murphy but he could only find the side-netting.
Stewart redeemed himself for that mistake three minutes later when he brilliantly denied Law after a neat through-ball by Murphy.
The home side were not to be denied for much longer though and they took the lead after eight minutes when Hateley's corner somehow evaded everybody inside the penalty area and bounced into the corner of Stewart's goal.
Morton were struggling to contain the home side's attacking threat and Ojamaa almost doubled their lead on 10 minutes when he fired
against the post after being released by Michael Higdon.
Motherwell grabbed the second goal their play deserved after 28 minutes.
Stewart produced a fantastic point-blank save from Higdon but Murphy was on hand to bundle home the rebound.
Murphy was proving to be a constant threat to the visitors' defence and he tested Stewart before Motherwell moved 3-0 ahead.
Shaun Hutchinson rose to bullet Hateley's corner beyond Stewart and leave the visitors with a mountain to climb.
The task facing Morton was made even greater on 34 minutes when the hosts stretched their lead to four.
Ojamaa twisted and turned inside the box and unleashed a low effort into the bottom right corner of the Morton goal.
Estonian striker Ojamaa turned provider for Motherwell's fifth goal, sliding a perfect pass in to Nicky Law who coolly dispatched the ball beyond Stewart.
Morton finally managed their first attempt on goal after 56 minutes, when Peter MacDonald did well to get his shot away, but Darren Randolph in the Motherwell net gathered easily.
Michael Tidser was next to test Randolph as the visitors enjoyed their best spell of the match.
Stuart McCall introduced Stuart Carswell and Humphrey in place of Law and Lasley as the home side looked to rest some tired legs and the introduction of the latter paid off quickly.
The winger's blistering run ended with a cut-back to Ojamaa who fed Murphy to lash home the sixth goal.
With six minutes remaining Higdon passed up a good opportunity to net his first goal of the afternoon when he glanced a close-range header over the bar but it mattered little as Motherwell's spot in the quarter-finals had already been long secured.