The Watford loanee slotted home after a penalty-box scramble to cancel out Zac Thompson's opener and give the visitors a barely-deserved point.
Bury had dominated the first half, Matt Doherty and Dominic Poleon both going close early on.
The Shakers eventually went in front nine minutes before the break, Poleon feeding fellow Leeds loanee Thompson, who rifled an inch-perfect finish into the bottom corner of the net.
Defender Mark Hughes then wasted a chance to double the lead, heading wide after he had stooped to meet Steven Schumacher's corner while David Worrall flashed a dangerous effort over the bar.
Tom Hopper had a chance to wrap up the win in the final minute but was denied by goalkeeper Mark Gillespie after breaking clear. However, Garner broke Bury hearts, keeping a cool head when the ball broke to him late on to ensure the spoils were shared.