Milan raced into a 3-0 lead in little more than an hour with goals from Keisuke Honda, Sulley Muntari and a Jeremy Menez penalty.
But Lazio, who had more chances throughout the game, came back strongly after defender Alex, making his debut after a move from Paris St Germain, put through his own goal.
Lopez saved a stoppage-time penalty from Antonio Candreva as the former Real Madrid goalkeeper kept the lead intact.
Former Italy and Milan forward Inzaghi became the latest to occupy the hot seat after Massimiliano Allegri and Clarence Seedorf were both fired last season.
Milan completed the signing of Fernando Torres from Chelsea shortly before kickoff but their finishing was efficient enough without him.
In front of a disappointing San Siro crowd, they got off to an ideal start when Stephan El Shaarawy burst forward and found Honda who scored unmarked at the back post in the seventh minute.
Ghana midfielder Muntari was left completely unmarked as he turned in Ignazio Abete's cross from close range in the 56th minute, then Menez appeared to make the game safe as he won and converted a penalty eight minutes later.
But Lazio, playing their first competitive game under new coach Stefano Poli, suddenly came back into the game when Alex turned Candreva's cross beyind Lopez three minutes later.
Stefano Mauri clipped the crossbar as they continued to press, then Lopez made a superb close-range save to deny Candreva before saving the same player's weak penalty.