Przemyslaw Wisniewski and Salvatore Esposito struck their first Serie A goals in the final 15 minutes to give Spezia their first win in two months.
Leonardo Semplici's team are still in the relegation zone but Saturday's win moves them level on 30 points with Verona, who sit one place above the bottom three and host Torino on Sunday.
Next weekend Spezia travel to Lecce who are just two points above the drop zone in 16th following their 2-2 draw at Lazio on Friday night.
"It's a goal I've dreamed about since I was a child, I used to talk to a good friend of mine who is a big Milan fan and I'd always tell him that I'd score against them," said Esposito to DAZN.
"These are big points for Spezia... we never lost faith, otherwise you'd never see performances like the one today."
A largely even match turned in the hosts' favour in the 75th minute when Wisniewski pounced after Esposito's corner was headed against the post by Kelvin Amian.
And Esposito sealed a precious three points 10 minutes later with a stunning free-kick which left Milan goalkeeper Mike Maignan grasping at air.
Milan will try to overturn a 2-0 first leg deficit against Inter on Tuesday but on Saturday's evidence they will have their work cut out for them.
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