Palermo maintained their 100 percent home record with a 2-0 victory over Fiorentina and visiting Bologna held Cagliari to a 1-1 draw.
Nineteen-year-old Dutchman Castaignos struck when he collected Thiago Motta's incisive pass and placed his shot into the bottom corner.
It was his first league goal since his move from Feyenoord in the close season.
Siena had enjoyed the better of the game and their frustration boiled over two minutes later when Franco Brienza was sent off, receiving a second yellow card for kicking and breaking the corner flag after a decision went against him.
Brienza hit the crossbar in the first half while Inter defender Walter Samuel wasted a good chance by firing wide from six metres.
Inter's second successive win lifted Claudio Ranieri's injury-hit team to 12th with 14 points from 11 games.
Cesena moved off the bottom thanks to former Chelsea, Juventus and Fiorentina forward Mutu's double.
The Romanian converted a 70th-minute penalty and scored with a shot on the turn 10 minutes later.
Cesena, who have nine points, overtook Lecce who lost 1-0 at home to Catania on Saturday.
Goals by Fabrizio Miccoli and Josip Ilicic gave Palermo their sixth straight home win, taking the Sicilians to 19 points in fifth place.
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