The tie was feisty from the off with combative tackles flying in from both sides, but a pattern started to emerge where the hosts shaded the possession stats while Lazio looked dangerous on the break.
Boulaye Dia had the first sight of goal when his deflected effort looped just wide, while Gustav Isaksen threatened from distance on more than one occasion.
Despite looking comfortable, Lazio fell behind late in the half thanks to a piece of magic from Marko Arnautovic.
An Inter set-piece was well defended with the ball being cleared high into the air, but the Austrian forward showed the poise and class to smash a first-time volley from 30 yards out to stun Christos Mandas.
The break gave Lazio the chance to regroup and Isaksen continued to cause problems after the restart, but Josep Martinez was equal to the shot after a fine mazy run into the area.
The visitors continued to push for a leveller and did well at the back post to improvise a powerful shot on goal, but again it was straight at Martinez who was well-positioned to make the stop.
Just as it looked like the Biancocelesti were gearing up for an onslaught in the final 15 minutes, ex-Lazio man Joaquin Correa earned the hosts a spot-kick when he was clumsily felled in the box just as he was about to pull the trigger.
Hakan Calhanoglu was forced to wait to take it as the visitors protested, but he was unfazed as he calmly sent Mandas the wrong way to double the lead.
Marco Baroni’s side didn’t give up and almost got back into the game immediately, but their luck in front of goal was summed up as Pedro’s deflected effort clipped the bar on its way over.
Following that Inter were able to see the game out to secure the win, leaving Lazio to crash out at the quarter-final stage for the fifth time in their last six Coppa Italia campaigns.