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.
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.
