Wayne Rooney scored the only goal of the game and later missed a second-half spot-kick after Jordan Spence was adjudged to have handled Ryan Giggs' cross.
West Ham had an equally strong claim for a penalty when Rafael kneed the ball on to his own arm, but Dowd waved play on, leaving Allardyce seething.
He said: "There's no doubt about the difference between Rafael's handball and Jordan Spence's.
"Spence plays for West Ham and the away team, while Rafael plays for the home side at Old Trafford.
"With Rafael, the ball hits his hand, but no penalty is given. With Jordan Spence, the ball hits his hand and it's a penalty.
"You see it time and time again at Old Trafford. The ref did not see Rafael's handball because of his position. But he was in a worse position to see Jordan Spence's than Rafael's."
Victory saw United through to a fourth-round tie against Fulham and United veteran Giggs was relieved to progress.
"I'm pleased," he said. "It's the FA Cup and anything can happen. You saw that from some of the results last night, so I'm pleased to get through.
"When you have got players coming back from injury it can be tough sometimes. And we have to give credit to West Ham - they were tough to play against."