In an attritional, forward-dominated contest, the 21 year-old fly-half scored all his side's points by landing four penalties as England ended a decade of Championship misery in Dublin with a first Six Nations win in the Irish capital since 2003 -- the last year they won the Grand Slam and, later, the World Cup.
Ireland, who had Slam hopes of their own after beating defending champions Wales in Dublin last week, saw replacement fly-half Ronan O'Gara tie the match at 6-6 with two second half penalties after England led 6-0 at the break.
However, with 10 minutes left, O'Gara missed a penalty and that meant Ireland now needed a converted try to win the match.
But with England captain and openside flanker Chris Robshaw producing a man-of-the match display at the breakdown, it was the visitors who emerged victorious.
Defeat ensured there was no double celebration for Brian O'Driscoll after the Ireland great's wife, Amy, gave birth to he couple's first child, a daughter, earlier Sunday.
England coach Stuart Lancaster said that the tough conditions had made it a grim battle between the two sides.
"I think we had a good first half, but at the start of the second we had a couple of turnovers that put us under pressure," he said.
"I think we grew in stature towards the end of the game and deserved our win. We hadn't won here for a long time and we gone and did it. We will take it."
Farrell gave England an early lead with a second minute penalty.
Tempers flared in the 14th minute when Ireland prop Cian Healy's apparent use of the boot on England's Dan Cole at a ruck sparked a mass brawl.
Both packs managed to slow their opponents' ruck ball and neither side had a genuine chance of a try in the first half of a match featuring several candidates for this year's British and Irish Lions tour of Australia.
In such a tight contest, discipline was especially important, and Farrell punished Ireland for coming round the wrong side of a ruck with a superb penalty from nearly 50 metres to make it 6-0 to England in the 28th minute.
Ireland lost fly-half Jonathan Sexton with a hamstring pull just after the half-hour mark, although the wet conditions put a premium on the accurate kicking game for which O'Gara is renowned.
But when O'Gara, Ireland's most capped player, held on too long in the tackle, following good work by England defensive linchpin Brad Barritt, it gave Farell a long-range penalty chance on the stroke of half time.
However, his kick just went wide.
Early in the second half Ireland won a scrum penalty and O'Gara cut England's advantage in half.
Coach Lancaster stiffened his side's physical presence by bringing on centre Manu Tuilagi for Billy Twelvetrees and Courtney Lawes for second row Joe Launchbury in the 48th minute.
Tuilagi missed England's 38-18 Calcutta Cup win over Scotland last week with an ankle injury, allowing Twelvetrees to make his Test debut.
England, though, were a man down in the 57th minute when blindside flanker James Haskell, was yellow carded by French referee Jerome Garces for kicking the ball out of a ruck.
O'Gara landed the ensuing penalty to tie the match at 6 6 heading into the final quarter.
Yet despite being reduced to 14 men, England then outscored Ireland 6-3 in the 10 minutes Haskell was off the field.
Farrell's clever kick ahead set up an England lineout close to Ireland's line and then Tuilagi was just unable to get a touch for a try following a neat chip ahead by scrum-half Ben Youngs.
However, Garces had already awarded England a penalty from inside the 22 and Farrell made no mistake to nudge his side into a 9 6 lead.
And that became 12-6 when the composed Farrell landed his fourth penalty after Ireland infringed by not releasing.
Ireland then saw O'Gara miss an eminently kickable penalty from just outside the 22 and England's impressive defence held firm.