Mac Allister gives Albion the lead shortly before the break as he nods in from Jahanbakhsh's cross.
Jahanbakhsh doubles Albion's advantage with a stunning strike into the top left corner early on in the second half.
Bernardo heads home to make it 3-0 just before the hour mark, before Gyokeres adds a fourth from close range.
Albion through to the third round of the competition, where they face Preston North End next week at Deepdale.
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