The Black Cats took a further step towards the Premier League trapdoor after another sterile display at the Stadium of Light and despite a late rally led by Adam Johnson, the damage caused by an impressive Andy Carroll proved fatal.
But the Uruguayan maintains that his side are not going down without a fight and believes if they continue to play as they have done in recent matches, they still have a shot at safety.
"I try to be honest, that’s what we are all about – that’s it," Poyet told reporters after the game.“It hurts yes, it's sad yes, but if you see this team giving up before the end of the season you won’t see me here.
“If we play like we play against Liverpool and like we did today then we are going to have a great chance. What I believe is that I need to convince the players to keep going.”
A few eyebrows were raised when the influential Johnson was left out of the starting line-up once more, but the Black Cats manager was adamant he stands by his decision.
“The only down thing was Johnson,” he added. “The situation is here is that I’m picking the team and I’m responsible.
“When it was time to take a risk we took a risk. Can you play the whole game like the last 20 minutes? No it’s not possible.
“If I made a mistake picking the team – I’ll take it [the blame].”
West Ham boss Sam Allardyce, meanwhile, was thrilled as his side recorded back-to-back victories and he had nothing but praise for the manner of the victory, reserving special mention for the outstanding Carroll.
“I was very proud of the players,” said Allardyce. “In the Barclays Premier League it [finishing] is critical to whether you end winning the game of football and our start and our quality of ball in the box was impressive.
“It’s an outstanding victory for us and a very precious three points for us at this time in the season. It’s got us out of trouble and it’s got us almost safe.
“It’s a bit scary to think where we might be if we had him [Carroll] all season.
“There’s a quality with the size and stature of the man that it’s not just a heading ability – there’s an all round quality to the man. He’s quicker than he looks, super in the air and he’s young.
“If he continues to score it’ll be difficult for Roy [Hodgson] to ignore him.”