Michail Antonio’s first-half strike secured all three points for the Hammers and left their former boss empty-handed in his final trip back to Upton Park.
It was a cruel result for the basement battling Black Cats, who missed a host of chances to equalise in the second half.
Jermain Defoe spurned two glorious opportunities against his old club, who was perhaps booed more by Hammers fans than Allardyce, while substitute Jack Rodwell fired straight at Adrian when it looked easier to score.
And Allardyce admitted: “You can’t create that many chances and not come away with a result.
“We are kicking ourselves, which is worrying, when you play that well and don’t get any points. We are making life extremely difficult for ourselves.
“We did enough to win, never mind draw. It must be that black cat we ran over.”
The build-up to the game centred around Allardyce, whose relationship with the West Ham supporters was strained to say the least during his four years in charge.
He may have anticipated a hot reception yet nothing more sinister than a polite smattering of applause greeted the 61-year-old as he took his seat in the opposition dug-out.
“Jermain Defoe got more stick than me I think,” he added. “But I had four very good years at this club. I’ve been a part of the history and part of them going to a new stadium as a Premier League club.
“To come here for the last time, it would have been nice to get a result.”
Both sides had hit the woodwork – Mark Noble saw a drive tipped onto the crossbar by Vito Mannone and Wahbi Khazri clipped the frame of West Ham’s goal – before Antonio struck after half an hour.
The there was no doubt from Hammers boss Slaven Bilic that his side fully warranted the three points.
“We deserved to win,” said the Croatian.
“We scored the goal, we hit the post twice and we could have had a penalty for a foul on Dimitri Payet.
“But more than a few times there were situations where we tried to be too cute. This is the Premier League. Until you score the second one there is always the danger you will concede.
“But did they deserve to win more than us? No way.”