Schumacher had enough chances for a hat-trick as Swindon looked lost without the influence of departed manager Paolo di Canio.
Lowly Bury had no trouble containing the sporadic home attacks after Schumacher had opened the scoring in the 21st minute with a near-post strike from Joe Skarz's cross.
Schumacher fired over after 73 minutes and came close to adding to his 10 goals for the season in the last minute only to be denied by a fine save from Wes Foderingham.
Swindon had to wait until the 79th minute before managing their first shot on target, a free-kick from substitute Luke Rooney which caused no trouble for goalkeeper Trevor Carson.
Swindon's unexpected defeat was their first in three months while Bury's win gives them renewed hope of avoiding the drop from League One.