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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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.