Innocent Emeghara opened the scoring for the hosts on 21 minutes, before Antonio Cassano replied almost immediately.
But Siena took the lead for a second time with 24 minutes played and a penalty from Alessandro Rosina following a rash challenge from Cristian Chivu, which resulted in the defender being sent off, sealed the victory for the hosts.
"I’ve never been one to look for excuses and I’m not going to start now; it was an ugly defeat, full stop," he told reporters after the game.
“We’re all at fault but I’m the coach, so I can’t be the one to hide.
“But thanks to this strange season we’re still in the hunt. I for one am not going to hide. I’m convinced that we can achieve our goals just the same."
The defeat leaves Inter fourth in Serie A behind Lazio, Napoli and Inter.