The Nerazzurri had been one of three teams to take six points from a possible six at the start of the campaign, but a trip to the capital on Saturday (AEST) represented a first real test.
It was not one Simone Inzaghi's side passed at its coach's former home, despite initially recovering from Felipe Anderson's first-half header.
Lautaro Martinez took one of very few Inter opportunities to equalise, yet Romelu Lukaku had already been substituted by the time Luis Alberto restored Lazio's lead with a deflected strike, and there was no response second time around as Pedro added gloss to the scoreline with a late third.
Inter started the brighter of the two teams, but the first big chances fell to Ciro Immobile, who was uncharacteristically wasteful as he dragged one effort across the face of goal and tamely aimed another straight at Samir Handanovic.
Felipe Anderson's cutback created that second opening, and he was far more clinical five minutes before half-time.
Found by Sergej Milinkovic-Savic's gorgeous pass in behind, Felipe Anderson directed his downwards header beyond Handanovic for an opening goal that was deserved despite Inter's dominance in terms of the possession.
Martinez had Inter level when he reacted fastest to a loose ball in the Lazio area and prodded past Ivan Provedel, but the momentum waned.
Luis Alberto's blast benefited from a deflection off Nicolo Barella to leave Handanovic rooted and pick out the top-right corner, before Pedro cut inside from the left to curl beyond the goalkeeper after Immobile went down in the box.