Boa Morte wasted another decent chance for the visitors when his header from close range looped over the woodwork.
Luis Boa Morte and Pablo Barrera wasted good chances and Carlton Cole hit the post as the Hammers continued to press.
Meanwhile, winger Luis Boa Morte has held clear-the-air talks with Sullivan after criticising the co-owner on Sunday's Match of the Day.
Boa Morte was taken to hospital with his knee in a brace and further tests will determine whether he needs surgery.
West Ham dominated the first half, but the best chance fell to Luis Boa Morte, who mistimed a header from six yards.
The Newcastle defender's backpass was too short for Given enabling Radzinski to intercept and lay the ball back to Luis Boa Morte.
Indeed, former Fulham skipper Boa Morte could have, himself, ensured his old employers were defeated had he done better with two good chances.
Brian McBride broke the deadlock with an unstoppable shot before Barry Hayles added a tap-in, and Boa Morte's close-range effort sealed the win.
Then Boa Morte, under pressure from Brede Hangeland, put Cole clear through, but his right-footed shot was diverted to safety by Paul Konchesky.
And all game long he had overshadowed his team's new signings Luis Boa Morte - playing against his old club - and Nigel Quashie.
That led to Grant replacing Radoslav Kovac with Obinna after 64 minutes, and Valon Behrami came on for Luis Boa Morte eight minutes later.
But reports say Newcastle are ready to table another bid for Boa Morte who has expressed an interest in moving to St James' Park.
BBC: SPORT | Football | My Club | Newcastle United | Newcastle 'back in for Boa Morte'
Boa Morte hurt his right knee after catching his studs in the turf at the Beijing Workers Stadium following a tackle with Spurs' Benoit Assou-Ekotto.
The Hammers were frustrated in the first half as Carlton Cole saw a volley saved and another skim the bar, while Luis Boa Morte also went wide.
The home fans at the Ricoh Arena were in raptures, but West Ham were given a lifeline when Boa Morte pounced on Arjen de Zeeuw's weak clearing header.
Boa Morte was then guilty of a glaring miss when he put a free header wide from Bobby Petta's cross with a glancing touch in the 42nd minute.
In a statement on West Ham's website, the club said Boa Morte had called Sullivan on Monday to apologise for his comments, and that the matter was now resolved.
An unmarked Pablo Barrera nodded inexplicably wide from close range after being picked out by a Luis Boa Morte cross before another Nouble shot struck Noble and spun narrowly wide.
In March, Sullivan lambasted the squad in an open letter to supporters, a move which disappointed Boa Morte, who is out of contract and expects to leave the club this summer.
The Hammers were forced back by wave after wave of red attacks which came to very little before Luis Boa Morte blazed a shot high and wide from a Nigel Reo-Coker cross.
Hull's hopes of adding West Ham midfielder or striker Luis Boa Morte, also 31, to their squad are not proceeding so well with the two parties unable to agree on personal terms.
The Hammers failed to muster a single effort on target in the first half, when they were aggrieved that Turner escaped punishment after tangling with Boa Morte outside the box and Cole in it.
West Ham made the most of their one-man advantage as they piled on the pressure, and only a superbly timed goal-line clearance by Danny Higginbotham prevented Luis Boa Morte putting the home side ahead with a mishit shot.
"Mr Sullivan should have a bit more composure when he talks to the press to hammer the players, it's not helping anyone, " said Boa Morte, who scored on his return for the Hammers in Sunday's 1-1 draw with Manchester City.
Blackpool had to change their keeper at the interval with Ghanaian Richard Kingson replacing Matt Gilks and his first job was almost picking the ball out of the net as Portuguese midfielder Boa Morte latched on to a cute through-ball by Parker but shot narrowly wide of the right-hand post.
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