Bulgaria keeper Nikolay Mihaylov gifted the home side a sixth-minute lead when he failed to hold on to an easy cross from Russia captain Andrei Arshavin, allowing midfielder Roman Shirokov to fire into an empty net from close range.
"It's always good to start with a win," Advocaat told a news conference. "I know what needs to be done to improve our play. Today we played better in the second half and didn't give our opponents many chances to score."
Advocaat succeeded fellow Dutchman Guus Hiddink in May as Russia coach after the team missed out on this year's World Cup finals following a two-legged playoff loss to Slovenia.
The match - the final test for both countries before they start their Euro 2012 qualifiers next month - was played at a leisurely pace on a hot evening in St Petersburg.
The game was originally scheduled to be played in Moscow but was switched to St Petersburg because of health risks after acrid smog engulfed the Russian capital last week.
The Moscow region has been hit by a record heat wave, with temperatures reaching 40 degrees Celsius (104 degrees Fahrenheit), causing huge forest fires in the area.
Mihaylov kept the visitors from falling two goals down when he managed a fine kick to block Roman Pavlyuchenko's low drive midway through the second half after a great solo effort by fellow Russia substitute Alan Dzagoyev.
Mihaylov also did well to deny Konstantin Zyryanov's attempt from the edge of the box in the final 10 minutes.
The Bulgarians, missing top striker Dimitar Berbatov who quit the national team after they also failed to qualify for the World Cup in South Africa, rarely threatened Russia keeper Igor Akinfeyev.
Russia, top seed in Group B, begin their Euro 2012 qualifying campaign away to Andorra on September 3, while Bulgaria travel to London to take on England in Group G on the same day.
