The Russians, looking more like the side that reached the European Championship semi-finals two years ago and less like the team who have dropped almost 20 world ranking places since, outclassed Ireland but then almost threw victory away.
Advocaat's side withstood the late fightback to make the most of the Slovakia's earlier shock 3-1 defeat in Armenia and the boss said they wholly deserved to join their nearest rivals on six points after three games.
"Today Russia were the better side. The most important thing is that for the majority of the game the crowd saw an excellent Russia," Advocaat told a news conference.
"Now we are totally back in contention but we still have to play (against Macedonia) on Tuesday. If you get a bad result then this has been no good."
Ireland began the brighter and almost took the lead when Robbie Keane's cross-come-shot grazed the crossbar on eight minutes.
However Giovanni Trapattoni's entire back four were caught ball-watching two minutes later as a speculative cross rebounded to recalled Zenit St Petersburg frontman Alexander Kerzhakov to net a deflected opener from close range.
In-form Kerzhakov, who notched hat-tricks in domestic and European competition last month, then neatly stepped over a cross just before the half hour to allow 20-year-old Alan Dzagoyev to grab his first international goal.
JITTERY FINALE
Russia, who surprisingly lost 1-0 at home to Slovakia last time out, went 3-0 up when an effort from midfielder Roman Shirokov took a big deflection off home defender Richard Dunne.
Captain Keane pulled one back from the penalty spot on 72 minutes before Shane Long added a second goal for Ireland six minutes later to set up a jittery finale for Russia.
The Russians hung on and next travel to Skopje in four days time while Ireland - whose lack of adventure for 70-plus minutes cost them dear - now need a positive result in Slovakia on the same night to get their campaign back on track.
Trapattoni said a victory was not out of the question.
"Why not?" the Italian replied when asked if his side needed a win.
"It's important psychologically not to lose our confidence. We finished well tonight."
