Inter had been 2-0 up thanks to goals from Fredy Guarin and Rodrigo Palacio.
But a goal from Alessandro Diamanti and an equaliser from Manolo Gabbiadini sent the game to extra time, where Andrea Ranocchia won it for Inter in the very last moments.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.