Doyle's seventh-minute strike and McGeady's 20th-minute drive, which was deflected home by defender Ildefons Lima, were enough to secure a 2-0 victory in Andorra which was more comfortable than the scoreline suggests.
However, Russia's victory over Slovakia in Zilina means that three points against the Andorrans on Tuesday evening will assure them of top spot in Group B and automatic qualification for next summer's finals.
In the unlikely event that Dick Advocaat's men lose to the minnows, Ireland could overhaul them by completing a double over Armenia in Dublin, but the more realistic scenario is that a point will be enough to keep the Armenians at bay and send Giovanni Trapattoni's men into the play-offs with a trip to the finals in Poland and Ukraine at stake.
However, the Italian will be without left-back Stephen Ward at the Aviva Stadium after he picked up his second yellow card of the campaign, and with Kevin Kilbane injured, that presents the manager with a selection headache.
Ward's misfortune aside, Trapattoni will have been happy enough with what he saw at the tiny Estadi Comunal as the Republic recorded an eighth successive clean sheet.
Ireland arrived in Andorra la Vella knowing anything other than a routine victory could prove fatal to their hopes of making it to next summer's finals, and equally aware that even a maximum six-point haul from their final two games might only be good enough to book a play-off place.
In the event, the prospect of anything other than an away victory effectively vanished within 20 minutes as Trapattoni's men took the game by the scruff of the neck.
Andorra's part-timers might have been behind as early as the second minute had Lima not blocked Robbie Keane's shot, and keeper Josep Gomes was a relieved man two minutes later when, after he had spilled the Ireland skipper's initial effort, his follow-up crashed against the crossbar.
In the event, Gomes' clean sheet lasted just three minutes further.
Doyle unwittingly blocked Keane's goal-bound shot, but was alert enough to react first when John O'Shea headed the ball back into the danger area and claim his 10th senior international goal from close range.
The Republic extended their lead with 20 minutes gone as left winger McGeady, who gave full-back Alexandre Martinez a torrid time throughout, was the beneficiary of Glenn Whelan's clever free-kick, which completely wrong-footed the Andorran defence.
McGeady had time and space to drill across goal and although Gomes managed to make a partial block, the ball was deflected home by Lima.
Andorra offered little as an attacking threat, although central defender Emili Garcia had earlier headed over from a Sergio Moreno free-kick and Lima tested Given with a firmly-struck free-kick.
The only disappointment for Ireland was Ward's 31st-minute caution for diving and the wider consequences for both the player and his team.
Ireland returned knowing the points were all but secure, but equally aware that there was no room for complacency.
They would have killed off the game for certain had it not been for Gomes, who blocked Keane's 48th-minute effort and then tipped McGeady's stinging drive onto the post two minutes later.
Opposite number Shay Given had to make a regulation 59th-minute save from Alexandre Martinez after Sean St Ledger had misjudged a high ball, but Ireland should really have gone further ahead within two minutes.
Damien Duff forced his way into the penalty area and managed to bundle the ball through to Keane seven yards from goal.
The 31-year-old pivoted, but somehow managed to blaze high over with the goal at his mercy.
Trapattoni made changes as he attempted to conserve energy in the likes of Whelan, Doyle and Duff, but with Keith Fahey, Shane Long and Stephen Hunt freshening things up, the visitors continued to dominate.
O'Shea went close with a 74th-minute header from a Duff corner, heading just over the angle of post and crossbar, but Ireland were in a comfort zone.
Long headed just wide from Keane's 84th-minute cross as Ireland coasted in to the final whistle having ensured they remain masters of their own destiny.