Russia, top of Group A after a 4-1 demolition of the Czech Republic when Poland could only draw with Greece to start, looked as if they would build on Friday's victory after Alan Dzagoev headed home in the 37th minute.
However, the Poles levelled the score in the 57th minute, when captain Kuba Blaszczykowski scored after receiving a Ludovic Obraniak cross with a left-footed pile-driver.
"I thought Russia was better than Poland today, but obviously it was a very good game," the Dutch coach said at the National Stadium in Warsaw. "We played really well to get a good result. You know they (Poland) have support from 40,000 people, so they've got a big advantage. 1-1 was nice, but I thought the 1-0 was going to be it because we controlled the game."
Ahead of the match, Advocaat had insisted that Russia did not feel like top dogs on the back of their thrashing of the Czechs, and given that the Poles had managed to throw away an early lead to draw 1-1 against a ten-man Greece.
Asked if his squad had nonetheless let themselves get over confident against the Poles, he responded: "No, that's got nothing to do with it. The first game was 4-1. The second was like an away game and it was a beautiful goal to equalise.
Advocaat lauded striker and captain Andrey Arshavin, whose free-kick set up Dzagoev's goal, brushing off suggestions that he had appeared tired later in the game and should have been substituted. "He was our most dangerous player, he was dangerous, the sharpest with the ball, so I wasn't going to change him. Every time he got the ball, something happened, so I left him on the field.
He was pressed to discuss the importance of rising star Dzagoev, who scored a double against the Czechs. "I'm not going to talk about individual players. But he has a lot of talent, he scored another goal. He did that very well.
"We have to just make sure we qualify in the next game. Everyone is good, but we'll have to show that again on Saturday."