The match was preceded by a minute's applause for Phil O'Donnell, the former Celtic and Motherwell player who died after collapsing against Dundee United at Fir Park two weeks ago.
The home side were responsible for most of the action in the first half, but their first attempt on goal didn't arrive until the sixth minute when Scott McDonald's volley from the edge of the box was saved by Scott Christie.
Vennegoor of Hesselink fired wide from 16 yards out moments later, and Stirling's lone striker David McKenna fired high over the bar with an ambitious effort from 35 yards.
It was moslty one-way traffic towards the Stirling goal, but Vennegoor of Hesselink and Aiden McGeady's best efforts were continually thwarted.
The breakthrough came after Naylor reached the touchline for the first time in the match and fired a neat cross for the unmarked Vennegoor of Hesselink to head past the helpless Christie.
McDonald squandered a chance to double the lead just seven minutes into the second half, but the Australian international failed to connect properly with a Massimo Donati pass and his shot went wide of the far post.
A McGeady shot in the 57th minute finished up yards wide of the target, while an earlier mazy run moments lput Vennegoor of Hesselink through, only for the Dutchman to be flagged offside.
In the 70th minute Naylor provided a slick ball for McDonald to calmly side-foot past Christie and Celtic Park rose to its feet with relief.
Minutes later, Celtic keeper Artur Boruc was finalyl called in to actoin when he pulled off a terrific point-blank save from Albion defender Andy Graham, who had made himself available for Stuart Devine's incoming free-kick just six yards out.
Stirling were visibly tiring as the match wore on and just five minutes later Nakamura fired in the third from the edge of the box with a powerful left-footed drive to make it three and end Albion's Scottish Cup dreams for another season.
Celtic: Boruc, Hinkel, Caldwell, McManus, Naylor, Nakamura (O'Brien 75), Donati, Scott Brown (Hartley 76), McGeady, McDonald (Killen 82), Vennegoor of Hesselink. Subs Not Used: Mark Brown, O'Dea.
Booked: Scott Brown
Goals: Vennegoor of Hesselink 37, McDonald 70, Nakamura 75.
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