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.