The visitors had shocked their opponents by taking the lead after 67 minutes when shoddy home defending saw the unmarked Coady left with a simple chance to stab the ball into the net from 10 yards.
The O's fought back and deservedly levelled when Cox collected a pass from the right and sent a wonderful volley into the net from 20 yards.
Blades goalkeeper George Long had kept his side on level terms before the break with two excellent saves to deny Nathan Clarke and David Mooney had earlier squandered the best chance.
Having been put through by midfielder Cox, Orient's leading striker failed to find the target from 12 yards, screwing his effort hopelessly wide of the far post.
The visitors offered little in the first half but showed more commitment going forward after the interval as the quality of the game improved with Chris Porter extending Jamie Jones.
But it was left to Long to produce the last save of the match when clawing away a Mooney header deep into added time.