With a trip to Manchester City on the horizon, City boss Steve Bruce named a completely different starting line-up to the one that secured their first Premier League win of the season against Norwich on Saturday.
Those that came in hardly did themselves any favours in east London and only managed to progress from the second-round tie thanks to substitute Brady's touch of class in the additional period.
It had looked for the most part like Hull would get their just desserts for naming a reserve side against an Orient team who came into this game on the back of their best start to a season for 103 years.
The League One leaders' confidence was clear from the outset, with a Romain Vincelot volley creeping just over in the third minute.
The Os continued in the ascendancy and dominated a first half in which Hull failed to muster a single chance.
David Meyler saw a header cleared off the line shortly after the restart, although Orient fought back and had several chances to take the lead.
Yohann Lasimant and Vincelot saw goalbound efforts cleared late in the 90 minutes - wasted chances the hosts would rue.
Bruce brought on Brady for the second half of extra-time and the Republic of Ireland international had the desired impact, netting little more than a minute after his introduction to secure fortuitous progress for the Tigers.
Given the raft of Hull changes, it was no surprise that the hosts settled quickest.
Moses Odubajo won an early corner for Orient after darting into the box and Lloyd James' delivery was met by Vincelot, whose fizzing volley flew just over Steve Harper's crossbar.
The early pressure continued, with the east Londoners' width complemented by fine link-up play between David Mooney and Kevin Lisbie.
The latter nodded over in the ninth minute and Harper then had to sprint off his line to stop the former Charlton striker getting a free run on goal.
James and Mooney failed with hopeful long-range efforts as the Os continued to press for an opener, with Scott Cuthbert heading over under pressure.
Mooney then headed into the side-netting a swooping Cox ball from the left as Hull looked anything but a Premier League side.
It is not hard to imagine Bruce's half-time pep talk and his side responded well, with Meyler seeing his powerful header cleared off the line two minutes in.
Orient were quickly back in the ascendancy, though.
Joe Dudgeon miscued a clearance straight into Lisbie's path, with the former Manchester United full-back forced to throw his body in the way of the driven shot that quickly followed.
Odubajo and Mooney failed with attempts on goal of differing accuracy as Orient continued on top, although Bruce did not seem too concerned as he brought on youngsters Conor Townsend and Calaum Jahraldo-Martin for their Hull debuts.
James sent a free-kick wide as the hosts continued to threaten, with substitute Lasimant firing wide before seeing a goalbound header cleared away by Paul McShane.
Orient came closer still when Alex Bruce reacted well to prevent a Vincelot effort creeping over the line.
Hull showed late signs of life as former Orient striker Aaron McLean hit wide, before Jamie Jones denied Matty Fryatt with his feet.
Fryatt was again thwarted by the Orient goalkeeper, before referee Gavin Ward signalled extra-time.
There were no chances of note in the first period and Bruce chose to bring on Brady for the final 15 minutes.
It was a move that paid immediate dividends as the Manchester United academy graduated swivelled to strike home.
Cox forced Harper into a late full-stretch save but it was not to be.