However, his performance today may keep his hopes alive, as he made three excellent saves to prevent Chelsea from taking all the points on Sunday.
'I think we begged.
"The FA Cup is a massive trophy with a lot of history but it's not enough for us", the 28-year-old Spanish defender told BBC Match of the Day.
The visitors forged the first real chance of the game after Marko Arnautovic looked to bring down a looping delivery from the left, but any intention of a shot was blocked off by a stern defence.
A win would see West Ham complete a league double against Chelsea for the first time since 2002-03, a season in which the Hammers were relegated.
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Conte had conceded losing at home to Tottenham Hotspur for the first time in 28 years last weekend was a huge setback in Chelsea's bid to salvage a dismal campaign by qualifying for the Uefa Champions League. That said, their form has been so erratic over the past couple of months that in likelihood, they will welcome Liverpool to Stamford Bridge next month with the top four already beyond them.
Azpilicueta went there hungry to score a goal.
In the 51st minute, Moses was presented with a big chance to put Chelsea two goals up from Morata's pass but blasted over the bar.
Chelsea's hopes of qualifying for next season's Champions League hang by a thread after stuttering to a 1-1 draw at home to West Ham United on Sunday. David Moyes' squad has lost three out of its last four matches, but defeated Chelsea in December in will be looking to move further from the drop zone.
Defeat would be West Ham's 1,000th loss in the top flight.