Joe Shaugnessy had given the Reds a 22nd-minute lead while Cammy Smith added a second nine minutes before the break as Derek McInnes' team made a positive start to the third-round clash.
And they sealed their place in the League Cup quarter-finals with Vernon's treble as a comprehensive 5-0 success was wrapped up.
Aberdeen arrived at the Falkirk Stadium packing four successive clean sheets and the confidence boost that comes with sitting third in the Scottish Premiership table.
But their Championship opponents were in a positive frame of mind themselves having climbed to a similar position in their own competition following the 3-1 weekend win over Dundee.
It took the Dons' first penalty shoot-out success in over a decade to see off Alloa in the previous round but there was no need for such a tie-breaker this time after Shaughnessy's opener settled Aberdeen nerves after some evenly-matched early tussles.
A goal to the good, McInnes' men found the freedom to start stroking the ball about with some real verve, while Falkirk's disorganised defence did their best to assist their attacking efforts.
And it was little surprise that a second from Smith arrived nine minutes before the break after Shaugnessy was allowed to dart forward from right-back unchecked to add an assist to his earlier strike.
Two further goals in two inside the first 10 minutes of the second period from substitute Vernon put the game beyond doubt.
And Vernon, who started on the bench before replacing Calvin Zola after half an hour, claimed ownership of the match ball with a 73rd minute penalty as Aberdeen took up their place in Thursday's draw for the last eight.
Aberdeen have suffered their fair share of embarrassing cup defeats to Scotland's minnows in recent years and were anxious to kill off the threat of another mortifying exit early on.
However, the excitement was a bit much for striker Zola who was booked after slamming into grounded Bairns stopper Michael McGovern with a straight leg after just four minutes.
The former Bulton Albion frontman's next involvement almost proved to be just as painful for the home side as he flicked on Johnny Hayes' corner for Shaughnessy to direct at goal but Stephen Kingsley cleared the danger from under his own crossbar.
Falkirk were keen to threaten themselves and a quick counter saw Jay Fulton roll in Blair Alston but Jamie Langfield scrambled clear on nine minutes. The early tempo was frenetic and the visitors raced away from the resulting corner only to see Hayes denied by McGovern while one-on-one.
Irish defender Shaugnessy, however, was far more precise as he made the most of some non-existent marking to fire Aberdeen ahead.
Clark Robertson's throw in was aimed at Zola but his wayward first touch worked in 21-year-old Shaugnessy's favour as he collected the loose ball before whipping a low drive into the far corner of the net.
With his early booking leaving him on a tightrope, Zola was replaced by Vernon on the half-hour mark as McInnes decided his physical approach was too much of a risk to persist with.
Any lingering concerns the Pittodrie boss had, however, were settled six minutes later as Smith buried a low finish after Shaugnessy craned his neck to redirect Hayes' searching cross towards the back post into the teenage midfielder's path.
McGovern got a hand to the shot but Aberdeen in this mood were simply unstoppable.
The keeper did better, however, four minutes into the second half as he blocked from Vernon from Ryan Jack's cut-back.
But the Northern Irishman's resistance soon faded as Vernon struck twice in two minutes.
First he stabbed home at the back post after Robertson got a head onto Nicky Low's cross from the right and then, with Falkirk still reeling, the Englishman slammed into an empty net after McGovern was left sprawled out after saving an initial shot from Smith.
Bairns defender Will Vaulks' clumsy trip on Peter Pawlett 17 minutes from time handed Vernon the chance to complete his hat-trick from the spot and his emphatic finish high to the left completed a painful lesson for Falkirk.
- Sports & Recreation
- Cammy Smith
- Derek McInnes
- Calvin Zola