The hosts had looked like they would gain a vital point after Teemu Pukki cancelled out Ciro Immobile's header from the penalty spot in the 72nd minute.
Chelsea midfielder Jorginho had the final say, though, slotting home from 12 yards with 11 minutes remaining.
The goal means Italy have won all six of their Group J qualifiers and sit six points ahead of second-placed Finland and nine above Armenia in third.
#EuropeanQualifiers 🏆— Italy (@azzurri) September 8, 2019
A hard-fought win away in #Tampere thanks to #Immobile and #Jorginho goals means qualification from the @UEFAEURO finals is now just a step away! #VivoAzzurro #FINITA 🇫🇮🇮🇹 pic.twitter.com/nslvmvBBbJ
Finland started in positive fashion and almost went ahead in the fourth minute when Joona Toivio scooped narrowly over at the back post from a corner.
The hosts kept their more illustrious opponents at arm's length in the opening 30 minutes, although they were indebted to a wonderful save from Lukas Hradecky to keep out Stefano Sensi's 18-yard volley.
Immobile then went close for the visitors, glancing a header wide of Hradecky's far post from Federico Chiesa's teasing right-wing cross.
Italy continued to crank up the pressure as half-time approached, with Sensi flashing wide from distance and Hradecky blocking Chiesa's effort from an acute angle.
Mancini's men started the second half as they finished the first but had to wait until the 59th minute to break the deadlock.
Chiesa's cross from the right found Immobile at the back post and the Lazio striker made no mistake, thumping a header in off the underside of the crossbar.
Ciro Immobile has scored his first goal for Italy in 733 days 👏 pic.twitter.com/zT2nQ3itvN— ESPN FC (@ESPNFC) September 8, 2019
Finland improbably found a way back into the game 18 minutes from time when Pukki clipped home from the spot after he had ben tripped in the area by Sensi.
Italy responded well, though, and secured all three points when Jorginho beat Hradecky from 12 yards after Sauli Vaisanen was adjudged to have blocked Nicolo Barella's shot with his arm.