Goals from Yéremy Pino, Chris Richards, Jørgen Strand Larsen and Eddie Nketiah did the damage at Selhurst Park.
The result was striking given Poznań had scored in each of their previous 15 away matches in Europe, a run Pierre Sage's side ended with only their second clean sheet of the season.
Sage was pleased with the result and the manner of it.
"We are always happy when we win, but especially in this way of playing," he told Palace TV. "We scored four goals with a different way to score, with different players and a clean sheet."
Sage pointed to two spells where his side lost their grip on the game.
"There were two bad moments in the first half and in the second half, maybe 10, 15 minutes, we lost too much the ball at this moment and we have to be more consistent in all the game," he said, before praising his side's defensive discipline.
"Everybody wants entertainment, but sometimes entertainment is by defending together. It's more difficult for the players to understand this kind of thing, but I think this evening they did it and they did it well."
Sage singled out Pino, who scored, won a penalty and set up several teammates.
"He's not a world champion for nothing, you know," Sage said. "He's an incredible player when he's like this. That's why I let him on the pitch, because I think he enjoys it. He gives a lot of enjoyment to the fans."
Captain Adam Wharton, speaking to TNT Sports, struck a measured tone about a Crystal Palace side still finding its feet under a new manager.
"There's still a lot to work on, still early days, lots of new signings. We are trying to gel as fast as we can. It has been a slow start but tonight was a great step forward," he said.
"The manager is very attack minded, he wants us to get high up. In training we are working on those patterns and I think there were a few good moments where we played one-touch play and it is good to see that starting to gel."
Sage dedicated the win to the memory of former Palace goalkeeper Kevin Miller, who passed away earlier on Thursday, before turning to Sunday's trip to face unbeaten Leeds United.
"Big game," he said. "We have to be in our best shape at the moment and we have to keep the confidence we built this evening to get a good result there."