Six penalties from six attempts by fly-half Toby Flood gave England an 18-11 victory against an Italian side who forced England to hang on desperately for the last 15 minutes.
Lancaster said: "We're delighted to get the win and relieved because of the pressure they put us under in the last 15 minutes. We know we have to improve our performance going into the Wales game."
Lancaster now takes his team to the Millennium Stadium next Saturday to face Wales in a fascinating encounter, where a win would deliver England's first Grand Slam for a decade.
Welsh victory by seven points, however, would see Wales lift the championship for the second year running, so long as they stay ahead in the tournament try count between the countries.
He added: "Nobody is punching the air in the changing room. We know we can do better. We missed one or two opportunities but going in at half-time at 12-3 we thought we would be able to pull away and finish the job off. But we conceded a soft try and suddenly it was game on again.
"Italy grew and played well in that last 20. It's a case of lessons learned and we need to ensure we improve going into next week."
Italy, who had scrum-half Edoardo Gori sin-binned for 10 minutes for holding back Flood in the first half, fought back after the break with fly-half Luciano Orquera chipping to the wing for Luke McLean to touch down for the game's only try.
And while Flood extended England's lead again with two vital penalties, Lancaster's men were forced to hang on in the final quarter as Italy, for whom Orquera kicked two penalties, at times camped on their line.
England forwards coach Graeme Rowntree said: "The guys at the end were just hanging on. We have come through a massive scare. Italy delivered everything I expected and credit to our lads for keeping their composure and coming through that."