The Azzurri scored two tries in as many minutes at the end of the first half after mistakes from England full-back Ben Foden first let in Giovanbattista Venditti and then Tomasso Benvenuti.
Italy edged further ahead with a Kristopher Burton penalty before England fly-half Charlie Hodgson scored his second charge-down try in as many weeks.
Owen Farrell kicked England to victory with a perfect display, landing four penalties and a tough conversion.
"We are delighted to get the win. We started well but then had a mad five minutes and came in down at half-time," Lancaster said.
"The most pleasing thing from that point of view was there was no sense of panic."
Addressing what changed after the break, Lancaster said: "We felt there were one or two areas to work on, particularly trying to move the ball, get the ball to the 10 early and get more intensity and tempo in the game.
"We were pleased to do that in the second half and delighted by the contribution the bench made. I thought Lee Dickson and Ben Morgan made a great impact.
"Owen (Farrell) was fantastic in terms of his temperament and his ability to step up and take those kicks, which kept our scoreboard ticking on.
"We talked during the week about trying to increase the tempo and intensity in the game.
"I thought Italy played extremely well and put us under a lot of pressure. They were very difficult conditions to play in.
"Overall we are pleased to get the win and looking forward to getting to Twickenham to play Wales in two weeks' time.
"Our aim is to improve week on week. It was probably less than three weeks ago we met as a group and started training together.
"We are still building and learning as we go. There will be a lot to take out of this performance and review."
The best part of England's performance was in the scrum, where the likes of Alex Corbisiero and Dan Cole enjoyed impressive games against Italy's much-vaunted pack.
"The contest was everything we expected. We got some key possession and key penalties in the scrum," said England's forwards coach Graham Rowntree.
But Rowntree conceded England's first-half lineout wobbles and their breakdown work are areas that must be improved on before the Wales game.
"Overall I look at our breakdown and the lineout and we have lot to work on to give better ball to the backs," Rowntree said.
Hodgson is developing a nickname of 'Chargedown Charlie' after he blocked kicks from Scotland fly-half Dan Parks last week and, today, Andrea Masi, to score crucial tries.
"I just try my best to get to the kicker and put pressure on him. It is not so much about practicing, you just have to go for it and hope for the best," Hodgson said.
"To come back from 15-6 down and be successful shows what resolve we have."
Italy prop Martin Castrogiovanni suffered a fractured rib during the first half and is expected to miss the rest of the Six Nations championship.
"It will be difficult for him to recover in time," said coach Jacques Brunel.
Italy captain Sergio Parisse was frustrated his side could not close out a first Test victory over "a poor England side".
But he believes the Azzurri are moving in the right direction.
"I'm proud of the team and also of the fans who made it in this weather and stayed the whole game to encourage us. I'm upset for them, they deserved a positive result," Parisse said.
"We did a really good job. I'm not happy but satisfied, it wasn't easy today to play the ball as it was really slippy.
"I'm convinced that we're on a good path and we already got from last week and we're going forward. It annoys me to lose to a poor England side.
"We had the chances today to beat England as it wasn't at its best. We had a big chance, but now we have to move on and prepare for the next two away matches."
Italy missed eight points from a failed conversion and two dreadful penalty attempts late in the game from replacement Tobias Botes, kicks that could have won them the game.
"When the scoreline is very tight like this we have to become more intense and get more points with the boot," Brunel said.
"We lost some points again with the kicks but it's the first time for Botes I think and it's not easy. We'll grow during this tournament."
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