The Biancocelesti spurned an opportunity to regain their position in the precious Champions League places in Serie A after Alessandro Rosina and a double from Siena's Innocent Emeghara ensured Milan held onto the coveted third place on goal difference.
But the 65-year-old coach has accepted full responsibility for the loss.
"I take responsibility for the team and the tactics, but you have to compliment Siena for the way they played. There are no excuses for this defeat," he told reporters.
"We have to apologise to the fans because tonight they did not see the real Lazio. I did not need to raise my voice at the players – they are smart and know what they have done wrong."
Lazio host Borussia Monchengladbach in the Europa League on Thursday before resuming their league campaign on Monday by welcoming Pescara to the Stadio Olimpico.