Reports prior to the game suggested Inzaghi would be replaced as head coach if Milan, winless in five Serie A matches, failed to claim victory at San Siro on Tuesday.
And Lazio were able to pile on the pressure as Lucas Biglia's 38th-minute penalty proved decisive, despite Lorik Cana's dismissal at the end of the first half.
Biglia broke the deadlock from 12 yards, sending the goalkeeper the wrong way after his cross had been handled by Michelangelo Albertazzi.
Cana then picked up two yellow cards in the space of five minutes to offer Milan a lift, but Inzaghi's men were unable to find an equaliser, with Giampaolo Pazzini and Alessio Cerci seeing goals disallowed in the second period.
Cerci's strike was ruled out in stoppage time, with subsequent replays suggesting the Italy striker was offside by the narrowest of margins.
Milan were without Philippe Mexes, banned for four matches on Tuesday following his dismissal for grabbing Stefano Mauri by the throat in Saturday's league meeting between the sides, and Stephan El Shaarawy, who sustained a broken metatarsal in the same game.
Inzaghi made six changes in all, while Lazio's team showed five alterations, with Filip Djordjevic among those to drop out after fracturing his right ankle at the weekend.
Miroslav Klose spurned an excellent opportunity to put Lazio ahead in the ninth minute, shooting wide of the near post after intercepting a dreadful backpass from Ignazio Abate and taking the ball around Christian Abbiati.
With the visitors on top, it took a timely interception from Riccardo Montolivo to deny Marco Parolo a tap-in after Abdoulay Konko had headed across goal.
Milan eventually carved out a clear-cut chance after 33 minutes, Etrit Berisha reacting well to parry away a volley on the turn from Cerci.
Yet it was Lazio who moved ahead five minutes later. Albertazzi controlled a Biglia free-kick from the left using his arm and the latter sent Abbiati the wrong way with the resulting penalty.
Milan received a boost in the final minute of the first half, when Cana picked up his second yellow card in quick succession for a needless challenge from behind on Jeremy Menez.
Keisuke Honda, back from Asian Cup duty with Japan, was called upon to replace Sulley Muntari for Milan early in the second half, while Dusan Basta entered the fray after Mauricio - on his first start for Lazio - had been clattered by his own goalkeeper when defending a free-kick.
Another substitute, Diego Novaretti, made a vital intervention to cut out Abate's ball across the six-yard box before Pazzini was denied a leveller.
The forward found the top-left corner with a wonderful half-volley in the 68th minute, but was penalised due to the ball rolling down his left arm as he brought it under control.
Lazio - who will face either Inter or Napoli in the semi-finals - came close to a second goal with 12 minutes remaining, Keita Balde Diao striking the inside of Abbiati's left-hand post.
Milan were almost handed a lifeline when Cerci struck in injury time, but a marginal offside decision went in favour of Lazio to leave Inzaghi fighting to retain a role he only took on in the close-season.
