Albion trailed to first-half strikes from Anton Forrester and William Edjenguele but hit back late on.
Michael Symes reduced the deficit in the 83rd minute and McGurk levelled things up with his second goal of the season.
The Shakers deservedly took the lead in the 16th minute when Forrester received a Craig Jones pass and cleverly side-stepped two opponents before curling a superb shot past Dean Lyness from just inside the area.
Bury doubled their lead five minutes before the break when big defender Edjenguele out-jumped the Burton defence to head in a Jordan Sinnott corner.
Visiting goalkeeper Rob Lainton had his first save to make in the 50th minute when he brilliantly kept the ball out after Rene Howe got a wicked deflection to a Lee Bell shot.
Substitute Symes headed in a McGurk cross to set up a thrilling finale, which reached its climax in the 90th minute as McGurk’s shot just had enough power to carry through the hands of Lainton.