Liverpool's Luis Suarez opened the scoring for the South American visitors in the 48th minute but Russia substitute Alexander Kerzhakov levelled moments later in Moscow.
Russia next take on Lithuania on Tuesday night as their Euro 2012 preparations start heating up and Advocaat believes they are on the right track.
"We played against a very strong team, which ranks third in the FIFA rankings," the Dutchman said on www.rfs.ru.
"We can say that we are moving in the right direction. We know what we still need to add, and hope to do this before our first game at the European Championship 2012 against the Czech Republic."
Advocaat spent three years with Russia's domestic side Zenit St Petersburg from 2006 to 2009 before taking over from compatriot Guus Hiddink as the national team manager in 2010.
And the 64-year-old hopes to bring joy to the country by leading them to glory in Poland and Ukraine.
"I had three great years with Zenit, two good years in the national team of Russia," he added. "I'd like to perform successfully at the European Championship 2012, so that all were satisfied and happy."