The Azzurri, who had never beaten England in 17 previous Test matches, profited from two Ben Foden mistakes in as many minutes at the end of the first half to claim a 12-6 lead.
Wing Giovanbattista Venditti pounced on a loose ball to score in the corner and then centre Tommaso Benvenuti picked off a Foden pass on halfway to score under the posts.
Suddenly, a historic England defeat looked on the cards. A defeat that would have left interim coach Stuart Lancaster with tough questions to answer about his “new era”.
But Hodgson rescued England with his second charge-down try in as many weeks and Farrell completed a 14-point haul with four penalties and a tough conversion.
Italy still had their chances to win the game but replacement kicker Tobias Botes failed with two simple shots at goal, one which barely got off the ground.
There was a time when England registered 40-point wins over Italy as a matter of course.
True, they hammered them 59-13 at Twickenham last year - but not in Rome. Not any more.
England's last two wins in the Eternal City had been by five and four points and with Rome in the grip of its worst weather for half a century, the odds on this being any different were slim.
The snow stopped around 90 minutes before kick-off, allowing stadium groundsmen to repaint the lines red and use air blowers to clear much of the pitch.
England launched an early counter-attack, spreading the ball wide to David Strettle before Tom Palmer took it on and charged into the Italian 22, but Tom Croft was unable to keep hold of Dylan Hartley.
England eventually broke the deadlock with Farrell's first penalty after 26 minutes - but only after a mistake from referee Garces cost them a try.
When Sergio Parisse's pass went to ground, Strettle reacted quickest to toe the ball ahead but he went down after colliding with the retreating Italian fly-half Kristopher Burton.
Although Benvenuti swept up, he was clattered by Farrell and Phil Dowson dived on the loose ball to score - only for England to discover referee Garcon had not played advantage, when he should have.
It was debatable whether the collision between Strettle and Burton was a penalty at all but Farrell, who took time to find his range at Murrayfield last week, slotted the kick.
Farrell had his head bandaged and Italy lost Castrogiovanni, who was stooping as he left the field with an injury suffered in a tackle from Botha.
Lorenzo Citadini came on and immediately found himself under pressure from the England scrum. Italy were penalised and Farrell landed his second kick.
But England imploded with two mistakes in the space of two minutes to gift Italy the lead.
Foden could not hold onto a grubber kick, he collided with Ben Youngs and the ball fell to Venditti, who scored in the corner.
England's full-back then compounded his mistake by forcing an offload on the half-way line, which was picked off by Benvenuti who scorched under the posts for a converted try.
Burton missed with a drop-goal attempt but landed a penalty as England made an ill-disciplined start to the second half.
England had to respond - and they did so with Hodgson scoring his second charge-down try in as many weeks.
After scoring the decisive touchdown against Scotland last week, Hodgson blocked Andrea Masi's clearance from inside the Italian 22, regathered and scored.
Farrell landed a crucial conversion from wide on the left and then, after 54 minutes, a penalty from in front of the posts as England regained the lead.
England were in the ascendancy now. Hodgson almost sent Hartley careering through a half-gap and Barritt powered forward.
Dickson was caught at the back of a ruck but Barritt charged down the clearance on the Italy line but knocked on as he tried to gather the loose ball to score.
England's scrum delivered for them again, winning a penalty under the posts which Farrell landed with ease.
England conceded a penalty straight from the kick-off but replacement kicker Botes missed badly - and then scuffed another one embarrassingly after Chris Ashton had taken him out off the ball.
Had Botes succeeded with both, as he should have then England would have been trailing. As it was, they held out for another narrow victory.
PA
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