The visitors took the lead in the 42nd minute when David Worrall took on a Grella pass and slammed a right-foot shot inside the near post from 25 yards.
County equalised a minute after half-time when Bury's Giles Coke diverted Alan Judge's cross into his own net.
Bury responded almost immediately when Grella expertly chipped past Stuart Nelson to restore their advantage.
The away side stretched their midway through the second period when Coke teed the ball up for Grella, whose deflected shot deceived Nelson.
Sub Mike Edwards gave County hope in the 79th minutes when he planted a header into the top corner before Bury's Andrai Jones saw red after picking up two bookings. However his team made sure in stoppage time when Mark Carrington broke away to score with a sublime finish.
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