Ireland were already leading through Keith Andrews' 13th-minute header when central defender Andrei Stepanov was sent off for a second bookable offence, and by the time skipper Raio Piiroja joined him 13 minutes from time after being cautioned twice in quick succession, they were 3-0 down.
Jonathan Walters' 67th-minute header, his first senior international goal, justified manager Giovanni Trapattoni's decision to pick him, and Keane's late brace ensured Ireland will take a commanding lead into Tuesday night's second leg at the Aviva Stadium.
The home fans were furious with Hungarian referee Viktor Kassai for what they saw as his part in their side's demise, but Ireland made hay as they banished the bitter memories of five play-off defeats in six previous attempts to take a giant stride towards Poland and Ukraine.
Ireland's tale of woe has proved a source of intense frustration, and the injustice of the final chapter which cast Frenchman Thierry Henry in the role of villain two years ago left a bitter taste in the mouths of the nation.
There was a belief before kick-off, however, that this could be their time after they were seeded for the draw which handed them, on paper at least, the least taxing of the four options available, and home advantage in the second leg.
On the pitch, things could not have gone much better as Estonia initially showed they were not prepared to be cannon fodder for the Irish, yet ultimately contrived to extend their hospitality far too readily.
Tarmo Ruutli's men started brightly enough, but found themselves behind with just 13 minutes gone when Aiden McGeady, who proved a threat throughout, burst into space on the left before floating the perfect cross into the middle for Andrews to head home his third senior international goal.
Walters, preferred to Simon Cox as Keane's strike partner for the night, might have doubled his side's lead five minutes later, but his shot on the turn after Sean St Ledger had headed down Andrews' free-kick flew high and wide.
Estonia responded as dangerman Konstantin Vassiljev, whose five goals proved so valuable during their Group C campaign, looked threatening in the space between the Republic's midfield and defensive lines.
His viciously-struck 22nd-minute shot had goalkeeper Shay Given concerned enough to dive full-length across his goal, and the Aston Villa man was mightily relieved to see the ball fly just wide.
But as Vassiljev and Middlesbrough winger Tarmo Kink started to pin the visitors back, they left space at the back and Walters shot inches over from a tight angle after being played in by Glenn Whelan.
However, Estonia were dealt a major blow seconds later when Stepanov's night came to a premature conclusion.
The defender, who had been booked for an earlier challenge on McGeady, crudely up-ended Keane after being caught in possession, leaving Kassai little option but to double the dose and send him off.
Ruutli had to reorganise at the break and opted to field a back three in the second half with Kink and Dmitri Kruglov operating as wing-backs.
They might have fallen further behind within four minutes of the restart when Damien Duff fired in a skidding drive from a narrow angle, but goalkeeper Sergei Pareiko dealt with it comfortably.
Kassai incurred the wrath of the home fans four minutes later when he awarded a free-kick just outside the box rather than a penalty for a foul by Whelan on midfielder Martin Vunk.
Vassiljev's set-piece was deflected wide by Richard Dunne, and it was then that Ireland made their big push.
They put the game, and perhaps the tie, beyond Estonia inside four minutes when Walters headed home Keane's cross after Pareiko could only parry McGeady's 67th-minute shot, and the captain gleefully tapped home after the goalkeeper had blocked Andrews' 71st-minute free-kick with his feet.
Piiroja picked up a booking which would have ruled him out of the second leg anyway for a foul on Andrews, but he joined Stepanov back in the dressing room with 13 minutes remaining after being cautioned for a second time for handball.
The depth of the Estonian disaster grew two minutes from time when Keane converted a penalty after Ats Purje had tripped fellow substitute Stephen Hunt to leave the former Soviet republic's hopes in tatters.