Wayne Rooney's header via his shoulder from a Steven Gerrard corner just after the half-hour mark put the visitors ahead but defender Kamil Glik levelled for the Poles, also heading home a corner with 20 minutes remaining.
"It's my fault. That's cost us three points," England goalkeeper Joe Hart told ITV television after failing to get any contact on Ludovic Obraniak's corner which was nodded in by Glik.
England, who stretched their unbeaten run against Poland to 15 matches since 1973, top the group on eight points from four matches, with Montenegro second on seven having played a game less and Poland third on five from three.
"It's a decent point but we know we can play better than that. I believe this point will be a good one at the end of the day, I don't think any team will get maximum points here," said England captain Gerrard.
The match had been scheduled for Tuesday but was postponed due to a waterlogged pitch when officials failed to close the national stadium's roof despite a heavy rainstorm, angering thousands of fans who were kept in the arena for hours.
Earlier on Wednesday, the England players and the FA said they would reimburse the ticket costs for supporters who stayed on for the match while the Polish FA said it would set up a refund process for those unable to attend.
"The fans were outstanding staying an extra day. Their fans were great coming too. We made something good out of it," said Hart.
The match was largely dour, with players in the warm-up visibly anxious about the sodden surface and the pitch cut up during the match and made fluent football difficult.
Poland tried hard to inflict a first defeat on the former World Cup winners in nearly 40 years and after Glik's equaliser the momentum was with the home side but England held out.
"I feel disappointed. We had a chance. We could have won. But congratulations are in order to the team anyway," said Polish captain Jakub Blaszczykowski who missed the game through injury.
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