Bromby's 25-yard screamer put Leeds 3-2 up in the 87th minute, before Beckford's second goal of the game in stoppage time wrapped up victory.
It looked ominous for Stockport as Leeds went in front in just the second minute. Robert Snodgrass made a break from midfield before smashing home into the top corner from 20 yards.
Stockport struck back 10 minutes later, though, when Carl Baker curled in a low free-kick from just outside the box.
Stockport then shocked Leeds when they went in front on the hour mark. Baker crossed from the left and defender Johnny Mullins stooped to head home.
However, Leeds struck back six minutes later as Beckford headed home his 18th goal of the season from Patrick Kisnorbo's cross. Stockport winger David Poole was red-carded in the dying seconds following a lunge on Max Gradel.