Michail Antonio's first-half strike secured three points for the Hammers and left their former boss empty-handed on his return to Upton Park.
Never mind managing the Black Cats, Allardyce joked he must have run over one after his relegation battlers missed a host of chances to equalise in the second half.
Jermain Defoe spurned two glorious opportunities against his old club while substitute Jack Rodwell fired straight at Adrian when it looked easier to score.
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."