Keisuke Honda was among goals on his AC Milan debut, scoring the third with a tap-in on 47 minutes after a Robinho header and Giampaolo Pazzini's volley effectively ended the contest in the space of three first-half minutes.
Nicola Ferrari gave the boisterous 7,000 travelling Spezia fans something to cheer about with an stoppage-time consolation goal.
Seedorf arrived at the San Siro to applause from home fans to see his new club bounce back from the humiliating Serie A defeat to Sassuolo at the weekend that cost former coach Massimiliano Allegri his job.
Seedorf's Milan will host Udinese in the quarter finals next Wednesday knowing the Italian Cup could be their only route into Europe after a woeful start to the season.
They are currently in 11th place in Serie A, 10 points behind local rivals Inter Milan in fifth.
Napoli controversially beat Atalanta 3-1 to follow Milan into the last eight.
Giuseppe De Luca gave Atalanta a shock lead at the San Paolo stadium in the 13th minute when he pounced to tap in the opener at the end of a lightning quick counter attack, only for Jose Callejon to equalise a minute later with a stunning volley from a tight angle.
A chipped pass aimed at a clearly offside Gonzalo Higuain struck Christian Del Grosso on the back while he was looking towards the referee's assistant pleading for the flag to be raised and ricocheted to Insigne, who rolled home the second.
After the protests died down Callejon sealed a last eight clash with Lazio with 10 minutes left when he met Gokhan Inler's drilled long pass with another sublime volleyed finish.
Siena have a Tuscan derby with Fiorentina to look forward to after coming back from a fourth minute Sebastian Leto strike to hammer Serie A's bottom side Catania 4-1 in Sicily.
Goals from Michele Paolucci and Francesco Valiani put the Serie B side ahead before Nico Pulzetti and Valerio Rosseti struck in stoppage time to outrage the home fans in a sparsely populated Stadio Angelo Massimino.