Walter Mazzarri left San Siro frustrated after Inter made all the running but were unable to break down their resilient opponents.
The hosts were surprised early as Chievo took the lead with Alberto Paloschi's fourth league goal of the season, but they rarely threatened after his eighth-minute strike.
Yuto Nagatomo's fifth in the league ensured Inter were not behind long, but they were only able to create half chances as the hosts collected a fourth draw in seven Serie A games.
The draw lifts Chievo a place in the table and means they are now a point clear of the relegation zone at the halfway stage.
Both sides made two changes from their previous league outings, with Inter welcoming back Mateo Kovacic in place of the suspended Fredy Guarin and the visitors opting for Paloschi in attack instead of Sergio Pellissier.
Despite only taking one point from a possible 12 the visitors started well and broke the deadlock early on.
A move down the left ended with Hugo Campagnaro and Juan Jesus colliding in the Inter defence, the loose ball rolled to Paloschi inside the penalty area and he fired high above Samir Handanovic and into the back of the net from a tight angle.
Inter hit back almost immediately with a well-worked goal that Nagatomo converted from close range.
Ricky Alvarez teased the Perparim Hetemaj on the right before delivering a low cross into the six-yard box that the Japan international swept home.
Mazzarri's side should have gone into the break ahead but they saw another Nagatomo effort disallowed for offside, while Rodrigo Palacio and Alvarez both failed to force Christian Puggioni into a save with their chances.
Inter started the second half positively but were wasteful in attack as Kovacic, Alvarez and Palacio all spurned chances to put them ahead.
The latter came closest with a scooped effort following Alvarez's through ball, but the 31-year-old saw his chance loop onto the top of the goal with Puggioni untroubled.
Chievo did create an opportunity of their own in the 71st minute as Cyril Thereau and Paloschi broke on the counter attack, the French striker was played in by his team-mate but dragged his shot well wide of Handanovic's goal.
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