The 23-year-old struck 14 minutes after the break to send the Hammers into the fourth round courtesy of a deserved win in the North-East.
They took the lead with 35 minutes gone when Frederic Piquionne rounded Simon Mignolet to fire home.
The Black Cats were indebted to their Belgian goalkeeper when, having already denied both Obinna and Piquionne in quick succession, he kept out Scott Parker's effort from point-blank range.
Club record signing Asamoah Gyan briefly restored order with a 41st-minute header, his second goal in his first start for the club following a £13million-plus move from Rennes.
However, the Wearsiders were a shadow of the outfit which took a point from Arsenal at the weekend and they surrendered tamely in front of a disgruntled crowd of 21,907.
The Hammers, for whom skipper Parker was outstanding, will hope to repeat the feat at home to Tottenham on Saturday as they attempt to claim their first three-point haul of the campaign, while Sunderland head for Liverpool having seen an opportunity pass them by.
Ever since arriving at the Stadium of Light during the summer of last year, Steve Bruce has voiced the opinion that the club, while unlikely to win the Premier League title, is more than capable of lifting one of the domestic trophies.
For that reason, the Carling Cup is a priority, and he went into the tie insisting there was no reason the Black Cats could not make the latter stages of this season's competition.
In truth, they could have been out of it by half-time after a lukewarm opening 45 minutes during which the Hammers might have tied up victory.
Despite making seven changes to the side which claimed a first league point of the campaign at Stoke on Saturday, the visitors started brightly with Piquionne and Obinna causing problems, although initially without troubling Mignolet.
Sunderland, with nine of the men who were selected for Saturday's 1-1 draw with Arsenal retained and £13million-plus man Gyan added, initially struggled to get going.
However, they might have been in front with 15 minutes gone had Darren Bent not marginally overhit his cross to the unmarked Gyan, who just failed to reach it.
Nedum Onuoha, as he had done against the Gunners, headed a Jordan Henderson corner just over, and the home side were belatedly building a momentum.
But the Londoners were not content to soak up pressure and would have been ahead on the half-hour had Mignolet not palmed away Obinna's curling shot and then recovered well to claim Piquionne's follow-up.
The opening goal came 10 minutes before half-time, and it was the Belgian who had to pick the ball out of his net after Piquionne accepted Obinna's pass and calmly rounded him to score.
Mignolet was exposed once again two minutes later when Parker exchanged passes with Pablo Barrera and stabbed the ball towards the bottom corner, only for the goalkeeper to throw out a hand to turn away his effort.
But Sunderland responded with four minutes of the half remaining when Gyan glanced home another Henderson corner to spare his side a tongue-lashing at the break.
The might have gone behind once again within seconds of the restart when Barrera's shot took a wicked deflection off Cristian Riveros and dropped just wide of the post with Mignolet completely wrong-footed, although he was perhaps due a slice of good fortune after being beaten by Cesc Fabregas' outrageous fluke at the weekend.
Gyan fired wastefully over after being picked out by full-back Phil Bardsley following an enterprising run down the right four minutes later, but Barrera's 55th-minute chip was close enough to send Mignolet scurrying anxiously back towards his line before it dropped over.
But there was nothing he nor anyone else to do to keep Grant's men out four minutes later.
The impressive Parker combined with Barrera once again and when he slid a pass into the Mexican's run, he squared for Obinna to thump a low drive past the keeper and just inside the post.
Bruce immediately replaced the tiring Gyan with Steed Malbranque, who forced a vital save from keeper Marek Stech with six minutes remaining, and Mignolet had to deny substitute Carlton Cole deep into injury time as hopes of another late, late rescue act faded.