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Italy threatened to secure their first victory over England in 18 tests when they took a 15-6 lead early in the second half before flyhalf Hodgson's try started a fightback.
Farrell, who was flawless when offered a chance to kick at goal, added the other 14 points on his second test start.
"We recognise we still have a long way to go," Lancaster told the BBC. "The most pleasing thing was that there was no sense of panic."
Hodgson said England had kept their composure despite conceding two tries late in the first half.
"We felt we dominated in the first half but they got a lucky try," he said. "We kept plugging away, kept our heads and it came good in the end."
"To show the resolve we have from 15-6 down, to be successful in a place like this we've shown what we have."
A dour opening half on a pitch partially covered in snow burst into life when Italy, trailing to two Farrell penalties, stunned England with two tries in the final three minutes through Giovanbattista Venditti and Tommaso Benvenuti to lead 12-6 at the break.
Hodgson got England back into the game when he scored a try for the second week running after charging down a clearance and Farrell's boot then proved the difference as Lancaster's side followed up their opening 13-6 win over Scotland with another scrappy victory.
The match was the first in the Six Nations to be played at Rome's Stadio Olimpico, home of soccer's AS Roma and Lazio, but the conditions were hardly ideal for an expansive game.
England had begun with that intention, spinning the ball wide in two promising early attacks which floundered with poor handling, setting the tone for much of a first half punctuated by some aimless kicking and territorial exchange.
Farrell's first two penalties had nosed England ahead before the hosts, who lost battering ram prop Martin Castrogiovanni to a rib injury, exploded late in the half.
With Italy pressing, England failed to diffuse two kicks and, after the ball ricocheted off Ben Foden's chest, right winger Venditti gleefully pounced to dive over in the corner.
Kristopher Burton missed the conversion but the Italian fans who had braved the bitter cold and falling sleet were on their feet again just moments later when Foden's pass was intercepted by Benvenuti who burst clear to score under the posts.
Burton extended Italy's lead to 15-6 by squeezing over an early second-half penalty but minutes later the game turned after Hodgson's intervention.
The England flyhalf scored the only try of the game in the win over Scotland when he charged down and gathered Dan Parks' clearance and this time he got his hand's on Andrea Masi's hack before a kindly bounce gave him the chance to cross for his eighth try in 38 tests.
Farrell converted and then kicked another penalty to nose England ahead 16-15.
Another straightforward three points from in front of the posts from Farrell gave England breathing space, his unerring accuracy in direct contrast to Italy's replacement flyhalf Tobias Botes who was wayward with two penalties that could have put his side ahead.
The misses proved costly as England, like they had done away to Scotland, defended resolutely to the end.
England next face Wales at Twickenham on Feb. 25 while winless Italy travel to Dublin to face Ireland.