Needing to beat the European minnows to boost their hopes of a second-place finish in Group C, Keane slotted home after just five minutes to relieve any early tension on the night he eclipsed Shay Given as Republic's most experienced player.
Giovanni Trapattoni's men were frustrated in their attempts to double their advantage until Keane struck again ten minutes into the second half, and he completed his treble nine minutes from time to seal a comfortable win.
The game was only five minutes old when the impressive Wes Hoolahan and Jon Walters exchanged passes before the Norwich playmaker fed the ball out to the left, from where Aiden McGeady crossed for Keane to sweep a first-time left-foot shot past keeper Gunnar Nielsen.
The Aviva Stadium rose to acclaim a national hero before settling down in anticipation of the floodgates opening.
However, Trapattoni had warned during the build-up to the game that the Faroes would make life difficult for Ireland and while they were repeatedly at full-stretch, they managed to negotiate their way safely to the break without further mishap.
That feat was due in part to the efforts of Nielsen, who produced a string of creditable saves to rescue an at times porous defence, and the Republic's lack of a killer touch in front of goal.
Jon Walters might have made it 2-0 with just 12 minutes gone after being played in by Keane but as he unleashed his shot, full-back Jonhard Frederiksberg arrived from nowhere to make a vital block.
Keane might have doubled his tally with 26 minutes gone when he was presented with a one-on-one from a well-worked free-kick, but this time Nielsen came out on top.
Walters planted a firm header into the keeper's hands from Marc Wilson's cross seconds later, and Nielsen then tipped a rising effort from Keane over as the Republic pressed for a second goal.
He had to save twice from McGeady as the half drew to a close with Ireland well on top, but knowing they could not relax for a moment.
Hoolahan could have calmed any nerves within four minutes of the restart when he met another McGeady cross in front of goal, but he could not keep his header down.
But the game was effectively over with 55 minutes played when full-back Seamus Coleman, who had been picked out by Hoolahan, curled in a low cross from the right which evaded Walters' lunge but was promptly dispatched by Keane as he slid in behind him.
Central defender Atli Gregersen might have set up an interesting finale when he sent a 77th-minute header just over David Forde's bar, but Keane struck again with nine minutes remaining after McClean and Sammon had combined to present him with the simplest of chances.