Mouhamed Dabo and Lorenzo Paramatti saw red for Inter either side of half-time and they could only grab a consolation right at the death through Eloge Koffi Yao.
Chievo got their goals through Demetrio Steffe, Idriz Toksic and a double from Victor Da Silva.
Watch video highlights from the game below.
First plays third today in Group B as Atalanta take on AC Milan. You can find the latest score from that game via our Twitter feed here.
Siena 3-2 Cesena, Torino 3-2 Spezia, Sassuolo 1-1 Genoa, Sampdoria 4-0 Bologna, Modena 0-1 Carpi, Novara 0-1 Juventus, Empoli 1-0 Parma.
Cittadella 2-3 Cagliari, Udinese 2-0 Varese, Inter 1-4 Chievo, Padova 0-0 Brescia, Verona 3-0 Virtus Lanciano, Pescara 4-4 Ternana.
Bari 2-2 Roma, Crotone 1-3 Trapani, Napoli 4-1 Avellino, Palermo 0-0 Reggina, Latina 0-2 Fiorentina, Catania 1-0 Livorno, Lazio 4-0 Juve Stabia.