Fiorentina took the lead at the San Siro through Josip Ilicic's fierce penalty after Inter keeper Samir Handanovic tripped away striker Nikola Kalinic.
Kalinic slid in after Handanovic could not tip over Ilicic's shot, then converted Marcos Alonso's low cross.
Inter centre-back Miranda saw red for cleaning out Kalinic, leaving the home side to play the final hour with 10 men.
Roberto Mancini's men managed to reduce the deficit when skipper Mauro Icardi poked in his own rebound off the Fiorentina post, before former Blackburn striker Kalinic completed his hat-trick by calmly slotting in from close range.
Inter have not won the Scudetto since winning the Serie A, Champions League and Coppa Italia treble under Jose Mourinho in 2010.
But the Nerazzuri raised hopes of ending that barren run by winning their opening five league games.
Fiorentina began the match in third place after a fine start to the season under former QPR, Swansea and Leicester manager Paulo Sousa.
Sousa, who arrived from Swiss side Basel to replace Vincenzo Montella in the summer, has pledged to bring high-tempo football to a team which has finished fourth in each of the last three seasons.
And his philosophy of forcing the opposition into mistakes worked perfectly against Inter, who were torn apart by a series of defensive errors.
La Viola moved above their hosts on goal difference, with defending champions Juventus already 10 points behind after a sluggish start.
BBC Sport
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