Motherwell were left to rue missed opportunities after Duk's header gave Aberdeen a first win over them in six attempts.
The Dons twice took the lead through Bojan Miovski and Duk, with the latter also having a strike disallowed for offside.
Aberdeen being clinical with their chances was a crucial difference in this clash, and though Stuart McKinstry scored a fine equaliser, they missed good chances through Kevin van Veen and Blair Spittal.
There was some early VAR controversy as Aberdeen first had a goal allowed for Miovski, then disallowed for Duk.
That seemed to spark Motherwell into life and Kevin van Veen should have drawn them level, before Ross Tierney had a penalty denied for a challenge by Liam Scales.
Motherwell's Blair Spittal could only find the side-netting with a header, while Duk had a fierce shot well parried by Liam Kelly inside stoppage time.
Connor Shields coming on at half-time seemed to turn the tide for Motherwell and within 10 minutes he had fashioned an equaliser for McKinstry.
But slack defending, which was all too evident throughout, came back to haunt the Steelmen as Duk stooped to nod in the winner in 68 minutes.
Motherwell boss Stevie Hammell made one change to the side that lost 4-0 to Celtic in the Premier Sports Cup quarter-finals in midweek, with Ross Tierney coming in for Connor Shields.
His Aberdeen counterpart Jim Goodwin made three changes to the team that beat Partick Thistle 4-1 in the same competition. Jack MacKenzie, Connor Barron and Jayden Richardson came in for Hayden Coulson, Matty Kennedy and Vincente Besuijen.
A howler to play the ball straight to Aberdeen's Duk led to MacKenzie playing in Bojan Miovski, but Ricki Lamie saw out the danger.
The Dons took the lead in three minutes when Miovski was sent through on goal by Barron and he chipped the ball over outrushing Liam Kelly, was flagged offside immediately, but VAR overruled the decision.
Aberdeen appeared to double their lead in 11 minutes when MacKenzie's cross from the left picked out Duk, and his header was blocked by Kelly, and he pounced on the rebound to slip it past him and into the net, only for VAR this time to disallow for an offside.
Kevin van Veen had a great chance to level in 15 minutes when he rounded Kelle Roos, but slipped his angled shot just past the left post.
Aberdeen went close again in 23 minutes when Leighton Clarkson's cross from the left picked out Richardson on the right, but Lamie made a superb block to clear the danger.
Ross Tierney went down in the box in 30 minutes under a challenge by Liam Scales, but no penalty was given, with Aberdeen defenders reacting furiously.
Motherwell fans were on the feet in 31 minutes but Spittal's powerful header rippled the left side-netting instead of flying past Roos.
In first-half stoppage time Duk had a chance to double Aberdeen's lead, but Kelly was equal to it, pushing his angled shot away from the right corner.
Aberdeen should have doubled their lead in 48 minutes when Richardson crossed for Duk, but he headed wide of the post.
Duk went down under a challenge by Sondre Solholm in 52 minutes, but referee Euan Anderson wasn't interested, and VAR appeared to agree.
But Motherwell drew level in 55 minutes when sub Connor Shields darted down the right and picked out McKinstry, who swivelled in the box before lashing a shot into the bottom left corner of the net.
McKinstry had a hand as Motherwell nearly took the lead, playing the ball off of Spittal for van Veen, whose low shot was well held by Roos.
Aberdeen regained their lead in 68 minutes, however, when MacKenzie's cross from the left was met with a glancing header by Duk that flew past Kelly and into the near post corner.
Sub Dean Cornelius sent an angled shot well wide of the left post in 80 minutes as Motherwell tried to draw level.
Lamie went close with a header from Spittal's corner on the left, but it flew over.
There was late controversy as van Veen was decked inside the box in the final minute of stoppage time, but nothing was given.
MOTHERWELL: Kelly, McGinn (O'Donnell, 77), Solholm, Lamie, Penney, Goss, Slattery (Cornelius, 72), Spittal, Tierney (Shields, 46), McKinstry, van Veen.
Subs (not used): Oxborough (GK), Mugabi, Maguire, Morris, Miller, Ferrie.
Booked: Lamie (43), Spittal (53), Goss (90+1)
ABERDEEN: Roos, McCrorie, Stewart, Scales, MacKenzie, Clarkson (Besuijen, 83), Ramadani, Richardson, Duk (Kennedy, 77), Barron, Miovski (Watkins, 85).
Subs (not used): Lewis (GK), Morris, Duncan, Milne, Ramirez.
Booked: Duk (33), Miovski (52)
Referee: Euan Anderson.
Attendance: 5486.
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