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 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."
Midfielder Andrea Pirlo does not think Italy should be written off at the European Championship ahead of their friendly with Luxembourg at Il Tardini.
The Azzurri play the minnows and Russia before their highly anticipated opening group match against Spain on June 10.
And while Pirlo is aware Italy are not among the favourites to win Euro 2012, he is still confident they can spring a surprise.
"Our objective is to go all the way in this competition. We are not among the favourites and we know that, but I am convinced Italy can finish in the top four," he told reporters at a national team press conference in Florence.
"We enjoy playing our football together, have quality and hunger, so those are our strengths. We aim to reach at least the semi-finals, because we are Italy and we can do it."
Elsewhere, Bulgaria and Turkey will play each other at Stadion Salzburg in Austria, while FYR Macedonia face Angola as part of their preparations for a World Cup qualifying group which includes Wales and Scotland.