The result at a freezing, almost empty and hostile San Siro increased the pressure on coach Filippo Inzaghi, with Italian media having speculated before the match that defeat could lead to his downfall after six months in the hot seat.
Inzaghi, in his first senior coaching role, would become the third Milan coach to leave in a year if he loses his job.
Lazio beat Milan 3-1 in Serie A on Saturday and the home side's nerves were clearly on display as they gifted their opponents two chances early on.
A dreadful back pass by Ignazio Abete was intercepted by Lazio's Miroslav Klose but the former Germany forward shot into the side-netting after rounding goalkeeper Christian Abbiati.
Jeremy Menez then produced another shocking pass which gave Lazio's Keita Balde a clear run on goal but Abbiati rushed out of his area to block the danger with a risky sliding tackle.
Milan's luck ran out, however, when Michelangelo Albertazzi handled the ball in the area and Lazio's Argentina midfielder Lucas Biglia fired home from the penalty spot after 38 minutes.
Lazio then had defender Lorik Cana sent off for a second booking just before halftime but even that failed to revive the hosts.
Milan's Giampaolo Pazzini put the ball in the net in the second period but it was harshly disallowed for offside and Alessio Cerci had a goal chalked off for the same reason, though replays suggested he was also onside when he received the ball.
Apart from that, however, Milan, who have not won a game since the Christmas break, never looked like turning the match around.
Although Inzaghi is the obvious scapegoat, Milan's Ultras issued a statement on Monday blaming the club's problems on their transfer strategy and called for chief executive Adriano Galliani to be sacked.
(Writing by Brian Homewood; Editing by Ken Ferris)
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