Motherwell had one enforced change from Saturday's league win over Kilmarnock.
Suspended right-back Paul Quinn was replaced by Brian McLean and Northern Ireland international Stephen Craigan wore the captain's armband in Quinn's absence.
The game had been already postponed twice due to the Fir Park pitch being waterlogged.
And there was a further delay when Well keeper Graeme Smith was ordered by referee Eddie Smith to find another jersey before kick-off.
With Mickael Antoine-Curier and Colin McMenamin cup-tied for Dundee, Alex Rae introduced Bob Davidson and Derek Lyle, while Gavin Swankie came in for David O'Brien.
Dundee worked hard in the first half, limiting Motherwell's scope for attack, but too often the Steelmen were ponderous on the ball, allowing the Dens Park men to retain their shape.
David Clarkson had the best chance for Motherwell in the opening half.
From a quick free-kick he got in behind the Dundee defence, but his volley lacked the power to trouble Ludovic Roy.
Four minutes into the second half, Dundee took the lead through McHale, scoring only his second goal of the season.
Wigan manager Steve Bruce was at Fir Park to assess signing targets
The visitors had started the second half the more adventurous and when Smith failed to clear the danger around his Motherwell goal the ball fell to McHale.
He took one touch to control and another to lash it off the underside of the bar.
With confidence growing, Dundee doubled their lead.
McDonald broke down the left, delivered the ball early and although it was missed by Lyle, the unmarked Scott Robertson slid in at the back post to prod home from three yards.
Motherwell boss Mark McGhee responded to the two-goal deficit by bringing on Simon Lappin and Darren Smith for Steven McGarry and Keith Lasley respectively.
Just after the hour mark Smith headed into the net from a precise in-swinging Stevie Hammell corner.
On 82 minutes the impressive McDonald was sent off for a second bookable offence when he fouled Stephen Hughes from behind.
McDonald's Dens Park team-mate MacKenzie was also red-carded, referee Eddie Smith taking a dim view of the young defender's foul on Craigan.
Despite intense pressure from Motherwell, Rae's nine men held on to go through to face Queen of the South in Dumfries in the quarter-finals.
Motherwell: Graeme Smith, Brian McLean, Craigan, Reynolds, Hammell (Jamie Murphy 88), McGarry (Lappin 57), Lasley (Darren Smith 57), Hughes, Fitzpatrick, McCormack, Clarkson. Subs Not Used: Daniels, Danny Murphy.
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