Zanetti will now call time on his career, following 19 seasons with the San Siro club, after playing the full 90 minutes at the Stadio Marc'Antonio Bentegodi as Inter said farewell to a host of first-team regulars.
Esteban Cambiasso, Diego Milito and Walter Samuel are also set to leave, but it was former Inter striker Obinna who stole the show.
Ex-Chievo man Marco Andreolli sent Inter in ahead at the break, despite the hosts enjoying the better of the first half and striking the woodwork twice.
But Inter were reeled in when Nigeria international Obinna fired home from a tight angle before scoring the winner a minute from time to sour Zanetti's swansong.
Zanetti made only his fourth league start of the season in his final game before retirement, while there were also places in the Inter XI for outgoing pair Cambiasso and Milito.
Samuel was on the bench and Obinna was named among the substitutes for Chievo - who were already safe from relegation.
The hosts had the first sight on goal when Alberto Paloschi brought down a long ball but rolled it wide of the onrushing Juan Pablo Carrizo and his left-hand post.
Paloschi then crashed a header against the crossbar, before Danilo D'Ambrosio's jinking run and shot marked the first effort on target for an Inter side that had previously guaranteed qualification for the UEFA Europa League.
Chievo's Dejan Lazarevic was denied by the post just before the half-hour mark, but it was Inter who went ahead before the break thanks to Andreolli.
Following a free-kick, the defender struck a loose ball powerfully past Lorenzo Squizzi from inside the area to register his first goal for the club.
Inter were much improved after the break but failed to turn their possession into meaningful chances.
Obinna capitalised with 17 minutes remaining, pulling Chievo level with a low shot that Carrizo will perhaps feel he should have done better with at his near post.
There was more to come from Obinna, who duly secured all three points for his side with a drive from distance late on.
Focus will now turn to Inter's transfer plans for the summer. With plenty of incomings and outgoings anticipated, the team that begins their next campaign is likely to look vastly different from the one that finished Sunday's match.
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