Sunderland headed to Upton Park to take on a team who had managed just one win from their last eight Premier League matches, with the likes of Andy Carroll, Kevin Nolan and Stewart Downing all sidelined for Sam Allardyce's side.
But, despite enjoying the better of the chances, the visitors had to settle for a goalless draw, leaving them five points adrift of 17th-placed West Ham.
And manager Poyet lamented his side's inability to capitalise on the hosts' current problems.
"Coming here when there is no Carroll, no Nolan, no Downing and company, then this is the game you need to win," Poyet said.
"Because if you come here and they are on the pitch, then it is going to be a bit more difficult. I think we did everything to win it.
"I don't like the word 'luck' but we've conceded so many own goals I was waiting for a deflected shot to go in the top corner or a bad decision from the referee to give a penalty or whatever.
"We need to be perfect, unfortunately, to win the game."
Sunderland did at least gain a point on the two sides directly above them as both Fulham and Crystal Palace suffered defeats on Saturday.
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