In a topsy-turvy encounter at the Wast Hills training ground, United led 2-0, trailed 3-2 before levelling matters with eight minutes to go.
The real drama unfolded in stoppage time with Sam Vince saving a penalty and Birmingham netting the winning goal from the subsequent corner.
Boston (3-5-2): Sam Vince; Jordan Nuttell, Charlie Melville, Todd Devlin; Karolis Andriulionis, Miles Lynch, Charley Sanders, Jordan Sowerby, George Stainfield; Juris Kormis, Leon Squire.