And once again the Shakers showed an inability to defend a lead as Weatherstone headed in in the closing seconds.
The Shakers went ahead through Jon Newby in the 48th minute. Martyn Forrest dispossessed James Gould and Newby finished with a left foot shot for his fourth goal of the campaign.
Only Paul Bastock kept the Pilgrims in the match with a number of fine saves including a brilliant fingertip effort to deny Pavel Abbott in the 68th minute.
Boston, skippered by former Altrincham favourite Paul Ellinder, did not manage a shot on target until the 83rd minute when Glynn Garner produced a superb stop from a Mark Angel free kick.
And that looked to be good enough for Bury. But a team that allowed Swansea to come back from 2-0 down to 2-2 and Shrewsbury to recover from 4-0 to 4-3 did it again as Weatherstone popped up with his close range header.
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