Celtic 3 Stirling Albion 0
It was Celtic's first game since the tragic death of Phil O'Donnell - who won the Scottish Cup with them in 1995 - and a rousing minute's applause in honour of the former Motherwell captain echoed around Celtic Park just prior to kick-off.
Having won the Scottish Cup last year, Gordon Strachan expressed a strong desire to retain the trophy this season.
Strachan named a formidable starting eleven which included Shunsuke Nakamura, who returned from a knee injury, and Andreas Hinkel.
The German international full-back made his debut following a £1.9million move from Sevilla.
As expected, Celtic dominated possession from the very start as Stirling lined up in a 4-5-1 formation to try and soak up the pressure.
Scott McDonald was first to test Scott Christie in the Albion goal after six minutes as his 20-yard drive was parried to safety by the Stirling keeper.
Jan Vennegoor of Hesselink then came close to opening the scoring in the 12th minute, but he headed inches wide after a sublime Hinkel cross.
Celtic continued to enjoy the bulk of possession but rarely forced Christie into a notable save with Massimo Donati heading wide from a Nakamura corner in the 18th minute.
Eventually the holders took the lead in the 36th minute as Aiden McGeady's left-wing cross found the unmarked Vennegoor of Hesselink, whose diving header found the net from eight yards.
On the stroke of half time, McDonald had a fierce drive from a tight angle but Christie again denied the Australian with a fine save.
The one-way traffic continued after the interval and in the 51st minute Donati threaded the ball through to McDonald but the striker screwed horribly wide.
The quest for a second Celtic goal was becoming relentless by this stage and Scott Brown saw a wonderful half-volley sail inches wide of Christie's left-hand post.
However, Celtic keeper Artur Boruc was called into action in the 62nd minute but Steven Bell's long-range shot was comfortably held by the Pole.
In the 69th minute Celtic yet again passed up a good opportunity to score a second as Gary Caldwell headed over from close range.
A minute later the decisive second goal did arrive when Lee Naylor's rampaging run down the left took him to the byline where he pulled the ball back from McDonald, who finished clinically from 12 yards.
To their credit though, Stirling continued to fight and Andrew Graham's header was saved brilliantly by Boruc in the 72nd minute.
Three minutes later though, Celtic put the tie beyond any doubt as Nakamura took advantage of a luxurious amount of space on the edge of the Stirling box and the Japanese magician curved an unstoppable shot high into the net.
Celtic now look forward to what will now be a tougher test in the next round of the Scottish Cup as they will host either Airdrie or Kilmarnock.