Aberdeen fans would have feared a repeat of their worst-ever cup exit, to sixth-tier Darvel two years ago.
However, moments after Aidan Glavin superbly turned over a close-range Peter Ambrose header, fellow forward Morris gave the on-loan Kilmarnock goalkeeper no chance with a 15-yard, side-footed finish from Dante Polvara's cut-back.
That eased the visitors' jitters and Ambrose, Morris and Nicky Devlin all missed chances before the break.
Ambrose fired another drive low and wide and Morris curled an effort off the outside of the near post before the latter was denied by a good Glavin save.
Elgin's brave challenge ended when Glavin was unable to prevent Duk's header from a corner crossing the line.
The hosts were reduced to 10 men following defender Lyall Booth's second yellow card and Duk curled in a late third.