Mats Hummels gave Germany the lead in the eighth minute when he met Toni Kroos' out-swinging corner with a powerful header, but Ignazio Abata pulled one back 20 minutes later and the score remained level.
Benedikt Howedes saw his half-volley come back off the right-hand post with the last kick of the game as Germany's 18-year winless run against Italy continued.
But Lahm believes they did enough over the 90 minutes to emerge triumphant.
He said: "We could have easily won here. As we have seen over the entire game we were the better team, we had the better chances, especially the shot against the post in the last second.
"We have done many things well, but we could still be better."
Joachim Low marked his 100th match as Germany coach and overall he was satisfied, particularly with his defenders' performances.
He added: "I'm pleased with how the game went. We ran it on the whole. We have put in a good performance and good defensive acts. For his goal Mats Hummels has prevailed in his class, but the Italians have also made it (a) great (match)."
