The result piled the pressure on manager Walter Mazzarri whose side were booed off at the San Siro after a comical defensive mix-up with 17 minutes left allowed Kone Panagiotis to tap in the equaliser before they missed from the spot.
Icardi had put the hosts in front in the sixth and 63rd minutes only for a fizzing Jonathan Cristaldo shot in the 35th and Kone's gifted goal to leave the home fans smarting after Diego Milito's weak penalty was saved seven minutes from time.
Inter were given the spot kick when Rodrigo Palacio was clumsily brought down by Andrea Mantovani only for Gianluca Curci to easily bat aside substitute Milito's effort.
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They could even have lost but for an astonishing double save by Samir Handanovic to thwart Robert Acquafresca in added time.
Mazzarri's side have not won in four games and stay fifth on 50 points, three ahead of Parma, who host Napoli on Sunday (1845 GMT), still occupying the last Europa League qualifying spot.
Inter are only four points ahead of red-hot Atalanta, who face lowly Sassuolo on Sunday (1300 GMT), and Hellas Verona, who beat local rivals Chievo Verona 1-0 earlier on Saturday thanks to veteran striker Luca Toni's 16th goal of the season.
AS Roma got to Cagliari on Sunday looking to close the gap on leaders Juventus to five points before Antonio Conte's side host Livorno on Monday (1700 GMT).
In the early kick-off, veteran striker Luca Toni struck his 16th league goal this term for Hellas to secure a win that takes them closer to European football next season.
Toni, who will celebrate his 37th birthday in May, scuffed home the winner following indecision in the Chievo area to move his side up to eighth place.
Defeat for Chievo means they remain 16th on 27 points, two above third-from-bottom Livorno.
Former Bayern Munich, Fiorentina and Juve striker Toni has been a revelation since signing for Hellas in the close season and is now two goals behind Serie A top scorer Carlos Tevez.
He has scored more league goals in one season than any other striker in the history of the club and could even find himself in contention for a place in Cesare Prandelli's Italy World Cup squad if he carries on his goalscoring exploits.
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