Midfielder Shuker, who has also played for Macclesfield and had a loan spell with Rochdale, gave Rovers the lead with a delightful 21st minute curling free-kick which left Blackpool loan keeper Lewis Edge no chance on his home debut.
And despite the Shakers creating a string of decent chances, luck wasn't on their side as their League One opponents did just enough to progress.
An 88th minute tap-in from Jennings capped a frustrating evening for Chris Casper and his side as they went in search of a financially-rewarding cup run.
Bury skipper Dave Challinor was harshly adjudged to have handballed on the edge of his own area against his former club and, with a superb dead-ball strike, it was Shuker who gave the visitors the lead.
Casper was left to contemplate what might have been as former Stockport winger Kevin Ellison teed up Jennings to seal victory. Tweet

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