Goals from Keisuke Honda, Sulley Muntari and Jeremy Menez gave the Rossoneri a comfortable victory at San Siro, with Alex's own goal a mere consolation for the visitors
A legend at Milan, Inzaghi was appointed as Clarence Seedorf's replacement in June and the Italian is tasked with returning the club to its former glories after a dismal 2013-14 campaign.
Keisuke Honda gave Milan an early lead at San Siro, before second-half strikes from Sulley Muntari and new arrival Jeremy Menez ensured the home side began the new season in ideal fashion.
Alex, another new signing for Milan, turned the ball into his own net midway through the second half as Lazio reduced the deficit, but the visiting side's evening was summed up when Antonio Candreva won and then missed an injury-time penalty as Milan debutante Diego Lopez denied him with an excellent save
Like the host, Lazio is also under new management, but Stefano Pioli was unable to garner anything from his first Serie A match in charge.
The result means Lazio's long wait for a league win at San Siro goes in, with their last triumph coming back in 1989, when Paolo Maldini's own goal secured a 1-0 victory.
Inzaghi included three debutantes in his starting XI as Lopez, Alex and Menez all featured, while Lazio's new signings Dusan Basta, Stefan de Vrij and Marco Parolo made their league bows.
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Milan made a bright start and was ahead as early as the seventh minute courtesy of a fine counterattacking goal.
Stephan El Shaarawy outpaced De Vrij down the left side and delivered a low ball across the box to Honda, who finished first time through the legs of Etrit Berisha.
The remainder of the first half was less eventful than the all-action opening and Lazio threatened briefly in the 37th minute when a fine flowing move was brought to an end by Cristian Zapata's timely challenge on Basta.
Lazio made a positive start to the second period, Miroslav Klose seeing a shot blocked before Alex had to be alert to clear Basta's dangerous cross from the right.
De Vrij then fizzed a 30-yard strike narrowly over the crossbar, but Lazio's hopes of an equalizer were dashed as Muntari doubled Milan's lead in the 56th minute.
The Ghana international threw himself at Ignazio Abate's right-wing delivery and prodded the ball home from six yards.
Milan looked to have put the result beyond doubt eight minutes later when Menez converted a penalty after referee Nicola Rizzoli adjudged he had been felled by De Vrij inside the area.
Shortly afterwards, Alex deflected Candreva's cross into his own net under pressure from Lazio substitute Filip Djordjevic.
As Lazio pressed for a second, Lopez reacted smartly to keep out Candreva's powerful volley at his near post five minutes from time, before the winger was fouled in the area, leading to Rizzoli pointing to the spot.
However, Lopez made a superb save low down to his left to deny the Italy international and round off a positive start to the Inzaghi era.