Obinna's 118th-minute strike sealed the win after Manuel da Costa's 96th-minute goal had put the Hammers 2-1 in front.
Obinna scuffed a good opportunity wide when through on goal, and Steele kept out Barrera with two more instinctive saves.
West Ham do have several attacking options, however, including new signing Demba Ba, Carlton Cole, Frederic Piquionne, Victor Obinna and Freddie Sears.
West Ham grabbed gratefully at the let-off and the outstanding Scott Parker swept home a third from Obinna's cross after 75 minutes.
And he then put Obinna through with a magnificent slide-rule pass to wrap up the win with two of the 120 minutes remaining.
That led to Grant replacing Radoslav Kovac with Obinna after 64 minutes, and Valon Behrami came on for Luis Boa Morte eight minutes later.
After a gentleman's agreement, hat-trick hungry Obinna was handed the duties of putting the West Ham ahead, which he did with a well-placed kick.
However, no less than ten of their stars from the latest campaign, including internationals Vincent Enyeama and Obinna Nwaneri, are expected to leave this time.
Steele made fine saves from Zavon Hines and Frank Nouble before Victor Obinna blasted a volley wide and Nouble was again denied by the visiting keeper.
The hosts dominated the first half and were unlucky to be denied a penalty when Craig Cathcart appeared to use an arm to block Victor Obinna's volley.
Some fans may have missed Obinna's opening goal in the fourth minute as they arrived late after London's transport system did its best to hamper their journey.
Obinna Nwaneri, who stepped up to tuck away the decisive fifth penalty during the shootout, was also confident Enyimba can win the competition a record three times in succession.
Victor Obinna's first league goal of the season doubled the lead early in the second half, although they still had to survive a scare before sealing what was ultimately a comfortable win.
West Ham made them pay for the miss with 15 minutes remaining, Parker feeding Obinna before surging into the area to slide in the return pass, a goal his performance richly merited.
Understandably, Obinna did not celebrate his second, which started out as a cross from way out on the right but ended up flying over the head of Camp and into the goal.
Moyes admitted at the weekend that his squad was too thin to improve on their fifth place finish last season and there could be further arrivals with Nigeria striker Victor Obinna a target.
It was only a temporary reprieve, though, as Obinna added the second after 56 minutes, receiving Piquionne's pass in the area before firing a thumping left-foot shot across Al-Habsi and into the bottom corner.
When Obinna was sent off shortly afterwards for a ridiculous kick on Larsson it seemed as though West Ham's hopes of reaching Wembley, and in turn Grant's of keeping his job, lay in serious jeopardy.
Parker's goal was West Ham's first from open play in the league this season, but Grant was pleased to see his team creating chances, with striker Victor Obinna twice going close to scoring on his debut.
Just after the half-hour mark the visiting keeper did even better to get down to his left and keep out Obinna's fiercely struck right-foot shot as Birmingham's defence continued to look suspect in front of him.
Frederic Piquionne nodded wide from a corner by Barrera - on for the injured Dyer after just 15 minutes and Hammers boss Avram Grant was rueing his luck when referee Kevin Friend failed to award a penalty after defender Cathcart slid in and blocked Obinna's volley with his arm.
After the hosts had been pegged back to 1-1 and seen Victor Obinna sent off for a crazy kick out at Sebastian Larsson, Cole bagged the winner with only 10 minutes left - and it was a personal nightmare for Blues keeper Ben Foster, who let the striker's soft shot roll under his hand and trickle over the line.
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