Marco Silva's charges began brightly as Leighton Baines and Michael Keane both came close in the opening five minutes of the encounter, while Cenk Tosun was next to threaten as he fired first time just over the bar.
Normal service resumed as Everton continued to press but the Shakers took the lead against the run of play just before half time when a free kick was worked to Neil Danns, who scored with a smart finish.
Silva made nine changes at the break and Oumar Niasse twice came close to levelling the scores but for the woodwork, while Sandro exchanged passes with Jonjoe Kenny and burst into the area before rifling over the bar.
The Toffees' pressure eventually reaped dividends as the Premier League side drew level on 63 minutes, Niasse finally finding the back of the net after his earlier misses when he slotted past goalkeeper Joe Murphy on the rebound.
Everton finished the game strongly as more chances came their way but the fourth-tier side held firm for a second draw against Merseyside opposition in quick succession, following their weekend stalemate with Liverpool.
BURY: Murphy (Hudson 71'), Miller (Cooney 64'), McFadzean (Stokes 78'), Thompson, O'Connell (Shotton 57'), Adams (Mayor 57'), Dawson (Allardyce 81'), Dagnall (Dai 71'), Aimson (Williams 71'), Danns (Styles 57'), Omotayo (Archer 57')
EVERTON: Stekelenburg, Coleman (Kenny 45'), Baines (Robinson 45'), Jagielka (Williams 45'), Keane (Pennington 45'), Schneiderlin (Besic 45'), Davies (Connolly 45'), Dowell (Klaassen 45'), Mirallas (Vlasic 45'), Sandro (Martina 75'), Tosun (Niasse 45')