Co-hosts Poland await on Tuesday in their second Group A match and with Germany, Netherlands or Portugal probably lined up in the last eight, the Russians cannot rest on their laurels despite Alan Dzagoyev scoring twice and Andrei Arshavin looking back to his best.
"At 2-1 it could have been 2-2. There were moments when we could have been sharper," the ultra critical Advocaat told reporters.
"We weren't really in the game the first 13 minutes. We weren't very sharp. From that moment on we kind of controlled the game," he said of Dzagoyev's 15th-minute opener.
If Advocaat was really looking for problems to address, Alexander Kerzhakov's wastefulness in front of goal was a thorn in his own side before rival forward Roman Pavyluchenko scored a cracking fourth when he came on as substitute.
However, Dutchman Advocaat backed his starter.
"I would choose Kerzhakov once again. You make certain choices and you have to back those choices. The only thing he didn't do was score," the coach added.
The Czechs must go back to the drawing board after goalkeeper Petr Cech, whose last competitive game was as a hero in the Champions League final for Chelsea, was left a mere bystander by Russia's attacking flair.
Playmaker Tomas Rosicky was also outshone by Arshavin despite having enjoyed a recent renaissance at Arsenal while his former Russian team-mate was forced to seek solace on loan at Zenit St Petersburg after losing his club form.
"We started well. The first 15 minutes we had ball possession, good passes. But then we lost the ball... and they got into a fast break. The Russians are a good team," coach Michal Bilek said ahead of their next fixture against Greece.
