Derek McInnes' side struggled to beat Alloa on penalties in the last round but he took no chances as he named a strong side.
Dons' substitute Scott Vernon scored a second-half hat-trick but the result already looked beyond doubt by then as they proved too strong for their Championship opponents.
Calvin Zola twice went close early on as the visitors showed their intent, although Falkirk almost took the lead through Jay Fulton with an effort that was kept out by Jamie Langfield.
Aberdeen continued to press, though, and they scored the opener midway through the first half as Joe Shaughnessy fired home his first goal since December.
Shaughnessy then went close to a second but his volley was cleared off the line by Craig Sibbald.
Instead it was left to Cammy Smith to double the advantage, the forward drilling home from 12 yards after Ryan Jack had knocked down a cross from Hayes after 36 minutes.
Vernon, introduced for Zola after he picked up a booking, then took control in the second half.
The former Oldham and Blackpool player scored twice in as many minutes, the first coming as he slid in to convert a cross from Nicky Low that had been flicked on by Fraser Robertson after 54 minutes.
Vernon then added another straight afterwards, pouncing to convert the rebound after Smith had seen his initial effort saved by Thistle keeper Michael McGovern.
Vernon spurned the chance to secure a swift hat-trick as his shot across goal missed the target but he sealed his treble with 15 minutes on the clock as he confidently slotted home.