Gary Alexander’s 61st-minute penalty broke the deadlock in Saturday’s clash and the visitors looked on course for a third straight League One win when Bury left-back Joe Skarz was sent off in the 78th minute at Gigg Lane.
But John-Lewis, who came on immediately after Brentford’s opener, volleyed home in the 85th minute to avoid a second straight defeat.
Brentford had the better of a hard-fought first half, with Niall McGinn firing wide early on, while Cameron Belford palmed away a Sam Saunders header.
Following a scramble in the Bury box, Mark Carrington brought down Toumani Diagouraga and Alexander converted the spot-kick with aplomb.
But Skarz was shown a second yellow for a foul on Marcus Bean, who tested Belford moments later and that proved a key moment as after Sweeney’s free-kick was only half-cleared, John-Lewis’ close-range volley was too strong for Lee.