Robbie Keane hit the bar early on at the Aviva Stadium but for the next hour Russia dominated and deservedly went three up through Alexander Kerzhakov, Alan Dzagoev and Roman Shirokov.
A Keane penalty in the 72nd minute gave the hosts a lifeline and, when substitute Shane Long added a second six minutes later, it seemed the Republic could get something from the game, but Russia held on to join their opponents on six points in Group B.
Victory for Armenia over Slovakia earlier in the day gave the Republic the chance to take a stranglehold on the group, and it briefly looked like they might be able to do just that.
After Russia had enjoyed the majority of possession in the opening seven minutes, the home side suddenly had three chances in quick succession.
First Aiden McGeady, who has been impressive for new club Spartak Moscow, tested Igor Akinfeev with a shot the Russia keeper could not hold onto.
Captain Keane picked up the loose ball and his cross bounced off the bar with Kevin Doyle waiting at the far post before another ball into the box fell for McGeady, whose fizzing cross-shot evaded all his team-mates.
It took only another three minutes for the breakthrough to come, but, unfortunately for the home fans, it was Shay Given who had to pick the ball out of his net.
Richard Dunne gave away what appeared to be a soft free-kick for a foul on Shirokov in the left channel and, when the set-piece was delivered by captain Andrey Arshavin, it took a slight deflection before reaching Given, who could only push it out to Sergei Ignashevich.
His overhead kick back across the box sat up perfectly for Kerzhakov and he managed to get enough power on his shot to beat the massed ranks on the Republic line.
The ease with which the visitors were finding space in and around the box would certainly have worried Giovanni Trapattoni as first Shirokov shot over when he probably should have done better before Kerzhakov tested Given with a fierce strike.
The Republic were not learning their lesson, though, and they found themselves two down in the 28th minute.
Aleksandr Aniukov was given time and space on the right to send in a cross, and a lovely dummy by the dangerous Kerzhakov allowed Dzagoev to drill the ball beyond a helpless Given.
The hosts at least managed a couple of forays into the Russia box with the half coming to a close as first Doyle shot weakly at Akinfeev before Keane just failed to pick out his strike partner after a promising move.
But the visitors could also have added to their tally, with Kerzhakov and Arshavin shooting off target, and at the start of the second half they were at it again as Aniukov found the side-netting.
McGeady, who has never scored for the Republic, then drew a decent save from Akinfeev but any hopes the home side had of getting something from the game seemed over two minutes later when Shirokov scored Russia's third.
There did not look much danger when the midfielder let fly from 25 yards but, in attempting to block the shot, Dunne deflected the ball into his own net with Given stranded.
The defender tried to make up for his error straight away, testing Akinfeev from distance, but Russia looked dangerous with every attack and Sean St Ledger needed to be alert to keep out Arshavin before Kerzhakov beat Given to the ball only to discover none of his team-mates had kept up with him.
The Republic's final change saw Keith Fahey introduced for Doyle in the 71st minute, and moments later they were given a lifeline when Keane chased a long ball into the area and tumbled under pressure from Chelsea defender Yuri Zhirkov.
There looked to be minimal contact but Dutch referee Kevin Blom pointed to the penalty spot and Keane sent Akinfeev the wrong way to extend his record goal tally to 45.
It seemed certain to be only a consolation but suddenly the Republic looked a threat, and they could have pulled another goal back had Kevin Kilbane's control been better six yards out.
Incredibly, given the one-sided nature of the match thus far, they did manage a second with 12 minutes still to play. Akinfeev saved again from McGeady but this time Long was following up and he managed to poke the ball just inside the far post.
The roar around the stadium was deafening but the hosts could not find an equaliser, and in fact should have conceded a fourth only for substitute Pavel Pogrebniak to fluff his lines six yards out.
The final action saw the home side screaming for a penalty when St Ledger went over in the box, but Blom blew his whistle only to signal the end of the game.