West Ham are 17th with 34 points while 18th-placed Hull remained on 28 with two matches remaining and a vastly inferior goal difference making their survival only the slimmest of mathematical possibilities.
Hull and Sunderland were reduced to 10 men just seconds before the break at the KC Stadium, with Jimmy Bullard missing a penalty for the hosts in the 41st minute of a bad-tempered match after Darren Bent scored in the seventh.
Hull's U.S. striker Jozy Altidore was shown the red card for apparently head-butting Sunderland defender Alan Hutton, who had thrown the ball at him after a touchline clash.
"We've been in the bottom three all season really so at the end of the day you've just got to hold your hands up and say we weren't good enough," Hull chairman Adam Pearson told Sky Sports television.
"We will take it on the chin, play out the next two games and make some tough decisions over the next couple of weeks," said Pearson.
"Administration is definitely not on the agenda, there are other ways forward which are slightly less scary which we will look at first."
"The boys really gave everything," said relieved manager Gianfranco Zola.
