The centre back nodded in an Andrea Pirlo free kick from the left to put Milan ahead in the 22nd minute with his maiden Serie A goal.
His compatriot Pato also used his head to good effect to put away a Ronaldinho cross and make it 2-0 10 minutes before the break before Silva unluckily deflected a Mauro Zarate shot into his own net in the 65th minute to set up a nervy climax.
The visitors held out under pressure to continue their recovery from a shaky start with a win that gives them 22 points from 12 games.
They are two points behind second-placed Juventus, who won 5-2 at Atalanta on Saturday, and seven adrift of champions Inter Milan, who drew 1-1 with AS Roma.
Lazio coach Davide Ballardini is now in serious danger of losing his job as his side have not won in the league since August and were hammered 4-1 at Villarreal in the Europa League on Thursday.
Milan leapfrogged Sampdoria, who lost 2-0 at Cagliari after having Marius Stankevicius sent off for a foul at the end of the first half.
Samp share fourth place with Fiorentina after the Tuscan team beat Udinese 1-0 away thanks to Juan Vargas's free kick six minutes from time.
