Allardyce: The big calls went against West Ham
Eden Hazard's 22nd-minute header was enough to settle an entertaining London derby at Upton Park.
West Ham had chances to earn a point, though, Diafra Sakho wasting arguably the hosts' best opportunity when he put a free header straight at Thibaut Courtois in the first half after Hazard's opener.
Manager Allardyce conceded his side should have done better with the opportunities they created, but felt Hazard was narrowly offside for his goal, while stating his team could have had a penalty when substitute Nene saw a shot hit the arm of Gary Cahill in the closing stages.
"A little bit of poor finishing and good goalkeeping has denied us," he said. "I've looked at the goal and it's offside.
"The reason I say it's offside is because Kevin Nolan had one ruled out at Old Trafford earlier this season to make it 2-2.
"There was half an inch of his nose offside and it was offside. There's about six inches of his [Hazard's] body offside and they said it's onside."
On the Cahill incident he added to BBC Sport: "I'm not sure it's a penalty, but you've seen them given. When it hits Cahill in the arm his arm wasn't in the natural position."
Allardyce, who confirmed that a hamstring injury was the reason for Winston Reid's early withdrawal, has now seen his team fail to win their past six Premier League matches.
However, the former Bolton Wanderers manager believes it will not be long before his team turn the corner.
"It was a good performance, you couldn't ask for any more entertainment," he added. "It was a terrific game of football, a fabulous effort from both players. We were on the wrong end of the result, sadly.
"We created so many chances against Chelsea, which is difficult to do because of how good they are not just offensively but defensively.
"So we pick ourselves up [and] if we play like that we'll start winning again."