Inter's decision to sack coach Stefano Pioli on Tuesday failed to galvanise the team who have failed to win any of their last eight league games and taken only two points during that run.
Pietro Iemmello scored both goals for Sassuolo as Inter dropped to eighth in the table, two places and four points adrift of the last Europa League place with two games left to play.
Pioli was dismissed on Tuesday and replaced on a caretaker basis by youth team coach Stefano Vecchi, who became the third occupant of the hot seat this season.
Pioli had been appointed in November after Frank de Boer was sacked after 85 days.
Inter, who have suffered a series of frustrating seasons since winning the treble under Jose Mourinho in 2010, were greeted by banners of protest from the Curva Nord as they entered the stadium, such as one which read "2016-17 season, let's be merciful and throw a veil over it."
After around 25 minutes, and with the score still goalless, a large number of fans walked out, leaving behind the banner: "You don't deserve our support, we came to say hello, now we're off to have lunch."
"We knew about the protests and the fans are right because they don't identify with this team," said Vecchi. "This team shouldn't be down in seventh and eighth place, so it's not surprising that they aren't happy."
"There is an atmosphere of great negativity which gets felt on the pitch," he added. "The slightest mistakes get punished."
Inter gifted Sassuolo the lead after 36 minutes when defender Jeison Murillo was caught in possession and Iemmello scored from the edge of the area.
It got worse for Inter when Iemmello tapped in from close range five minutes after the re-start.
Eder pulled one back with a deflected shot with 20 minutes left, briefly sparking Inter to life but it did not last and Sassuolo comfortably saw out the game.
"It's my fifth season here and it's always been the same story," said goalkeeper Samir Handanovic. "Inter cannot keep finishing seventh or eighth, their patience has run out."
(Writing by Brian Homewood in Berne; Editing by Toby Davis)
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