The striker has signed a two-and-a-half year deal with United, who are managed by former Rovers skipper Graeme Lee.
Bogle joined Rovers last January from Charlton Athletic. He was due to reach the end of his contract in the summer.
The 29-year-old departs Rovers having made 28 appearances, scoring three goals.
He was exiled by Richie Wellens at the end of August with the then-Rovers boss telling him he had no future at the club.
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Alongside Ed Williams he trained with the U18s under Gary McSheffrey, who then brought him back into the senior fold after taking charge following Wellens’ departure.
His last goal came in the defeat to his former club Wigan a fortnight ago while he came off the bench in the win over MK Dons last weekend.