The Royals, who remain in 19th position in the Premier League, took an early lead when Pogrebnyak intercepted James Collins’ poor back-pass and finished past Jussi Jaaskelainen.
The Hammers had chances to level, with Reading goalkeeper Adam Federici denying Collins before Ricardo Vaz Te missed a simple header from close range in the dying moments of the game.
“We shot ourselves in the foot from the early stages,” Allardyce told BBC Sport.
“James Collins has been solid for us this season but he made a basic error and they took full advantage. Reading defended deep in numbers and Vaz has missed a glaring, glaring chance and two errors mean we have not taken any points against them.
“We are slipping back down that way now and that is going to put pressure on us. We should have undone our early error but Vaz has missed it.
“We are bitterly disappointed not to come away with a point and now we are without James [Collins] and Kevin [Nolan] through suspension for an important game against Norwich.”
Reading manager Brian McDermott heaped praise on his side for producing a determined performance to secure just their second Premier League victory of the season.
“I’m very pleased,” McDermott added.
“We were very resilient today and we needed to be. We know what we have to do to get results in this league and we move on now and look forward to our next game against Spurs.
“We had to deal with facing a very physical side and we have kept another clean sheet. I didn’t feel they had too many opportunities. Pavel took his chance very well and we’ve also hit the bar.
“A couple of their players could have been sent off and you saw two teams desperate to win out there.”
MORE: Latest Man Utd news
MORE: Latest Arsenal news
MORE: The latest football news