Stoke midfielder Whelan, winning just his fourth cap after being thrust into the limelight by new coach Giovanni Trapattoni, fired a 70th-minute effort past bungling Georgia goalkeeper Giorgi Loria to seal a vital victory.
Ireland had taken a 13th-minute lead when Doyle headed home an Aiden McGeady cross, and could have been out of sight by half-time.
Georgia rallied briefly and should have been back on terms through Alexander Iashvili.
But after Robbie Keane had hit the post, Whelan made sure, despite Levan Kenia's injury-time strike, in front of a crowd of just 4,500 in Mainz to send his side into their trip to Montenegro on Wednesday in high spirits.
Much of the pre-match build-up had centred around FIFA's decision, prompted by the Football Association of Ireland's concerns, to switch the game to the neutral venue of FSV Mainz's Bruchweg Stadium.
The Georgians were not best impressed, with coach Hector Cuper insisting there was no reason why the tie should not have been played in Tbilisi, and the Argentinian's mood would not have improved when his side figuratively turned up half an hour late for their re-arranged date.
For 30 minutes, Ireland had things all their own way to the extent that goalkeeper Shay Given was a virtual spectator.
Most of the action came at the other end as Doyle marked his 20th senior cap with the opening goal 13 minutes into the campaign.
It came from the Republic's first cohesive attack, Steven Reid feeding the ball out to Keane on the left, the skipper in turn setting up Aiden McGeady to cross for the Reading striker to head past the flat-footed Giorgi Loria.
With Stephen Hunt - in for the injured Damien Duff in the only change to the side which drew 1-1 in Norway last month - and McGeady repeatedly swapping wings, Trapattoni's side dominated and might have increased their lead after 27 minutes.
Blackburn defender Zurab Khizanishvili's foul on Keane gave the Irish a free-kick 25 yards out, and when the skipper tapped it square to Steven Reid, he unleashed a stinging low drive which Loria did well to parry before claiming Doyle's looping header after he pounced on the rebound.
But just as it looked as though Ireland had assumed complete control, Georgia made their first big push and went in at the break wondering just how they were doing so still in arrears.
Dangerman Iashvili had earlier flashed two shots well wide of the target, but with eight minutes of the half remaining, he was presented with the most gilt-edged of opportunities.
Iashvili was left unmarked on the edge of the six-yard box as the Republic lined up to defend a Levan Kenia corner, but perhaps surprised at the time and space he had been given, he headed over the bar.
The lively Kenia proved a key figure for the Georgians as the game resumed, finding himself in the thick of the action at both ends of the pitch and testing the masked Kevin Kilbane's willingness to put his body to the test a week after fracturing his cheekbone.
With their lead looking slender, Trapattoni's side knew they needed some insurance, and they could have all but killed the game off within six minutes of the restart.
Defender John O'Shea met McGeady's corner unaccompanied by a white shirt, only to lift his header over the bar.
But Georgia were riding their luck and only just survived a self-inflicted wound with 57 minutes gone.
Keane could not believe his luck as he ran on to a desperately underhit back-pass to Loria and round the keeper and shoot towards goal from a tight angle.
To his despair, the ball came back off the upright and with defenders heading back in numbers to protect Loria's goal, McGeady fired a foot wide after the loose ball had been fed out to him.
But Kenia and Levan Kobiashvili, perhaps sensing that Georgia needed to make their move, started to pull the strings in midfield and but for Richard Dunne's 65th-minute block from Zurab Khizanishvili's shot, they might have been level.
However, Ireland's nerves were settled with 20 minutes remaining when Whelan capped off an expansive passing move by sending in a dipping shot which embarrassed Loria, who could only help the ball into his own net.
Kenia was rewarded for a fine individual display with an injury-time consolation strike, but it was not enough to deny the Irish their win.