Deegan netted in spectacular fashion in the 49th minute at Easter Road and with Williams saving Scott Vernon's spot-kick six minutes from time, the goal proved enough to take Hibs through to the quarter-finals.
After a turgid first period, Hibs took control of the contest immediately after the interval, with Irishman Deegan producing a dipping swerving drive from distance to break the deadlock for his first-ever Hibs goal.
Aberdeen, roared on by a vociferous following from the north, produced a late rally, with Russell Anderson, Scott Vernon, Niall McGinn and Peter Pawlett all coming close, but Williams was in incredible form.
Even better was to follow from the big Englishman as he saved Vernon's spot-kick - his fourth penalty save this season - after Jonathan Hayes had been clumsily felled by Hibs skipper James McPake.
An effort from distance from Alan Maybury which skewed wide was the sum-total of the first-half action, although Clark Robertson flashed a similarly wayward drive over the bar.
Hibs emerged in the second half with far more attacking impetus and within minutes tested Jamie Langfield for the first-time, with Scotland international Leigh Griffiths whipping in a fine free-kick from out wide which had to be clawed from underneath the bar by the Dons goalkeeper.
The resulting David Wotherspoon corner inadvertently struck Aberdeen full-back Robertson, and Hayes was forced to hook the ball off the line.
Hibs were cranking up the pressure as the contest finally sprung to life and, with 49 minutes on the clock, Deegan opened the scoring in stunning fashion, unleashing a ferocious right-footed shot from 30 yards which kissed the underside of the bar and nestled in the net.
Aberdeen finally found some attacking intent going into the final 20 minutes. Anderson, an unlikely source, fired in a good shot from distance which Williams had to parry over the bar.
Hibs introduced Matty Done for his debut but it was Dons replacement Vernon who almost made a telling contribution. The bustling striker latched on to an Anderson header across the face of goal but saw his effort from point-blank range clawed away.
Williams then made a laudable double-stop, saving a rasping Pawlett drive through a forest of legs with his feet, and managing to regain his balance swiftly enough to block Vernon's headed rebound.
And a man-of-the-match display was crowned on 84 minutes. Hayes' jinking run into the box was cut short by McPake with an industrial shoulder-charge, prompting referee Iain Brines to point to the spot.
Vernon stepped up but, once again, there was only one winner in that duel as Williams pushed his effort on to the post.
Aberdeen's last chance came through McGinn but his shot from the edge of the box was saved by the superb Hibs custodian, ensuring Pat Fenlon's men progressed to the last eight.