Rangnick's side took the lead through Gernot Trauner's ninth-minute header but allowed Poland back in when Krzysztof Piatek levelled after half an hour.
"If I'm really honest, we begged to concede this goal," grumbled Rangnick to reporters after the game.
Christoph Baumgartner and Marko Arnautovic netted after the break to help Austria take a step towards the last 16.
"After the 20th minute in the first half we dropped down in our performance level," Rangnick continued.
"We had a few players who underperformed in the first half compared to their normal level.
"We don't have an extraordinary squad that allows four or five players to be below par, but in the second half that changed completely."
Rangnick spoke in depth to Baumgartner at half-time and the RB Leipzig midfielder celebrated with the coach after sending Austria back in front.
"At half-time I told him, 'you're such an important player, we need you at least at normal level,'" said the former Manchester United interim coach.
Rangnick almost lost his spectacles in the effusive celebration.
"My glasses are still okay and I also have two extra pairs," he added.
Austria lost their opening game against tournament favourites France but held their own in a tight clash.
"We knew against France we weren't at our maximum (level), we know if we reach our maximum we can beat a lot of nations," said Baumgartner, named player of the match.
