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Italy made a fantastic start to the match when Antonio Candreva went down inside the box in the opening minutes, allowing De Rossi to convert the penalty at the second time of asking after being made to retake following suspected encroachment.
Italy's goalscorer almost put his side in trouble when he lost the ball in midfield, allowing Iliyan Mitsanski to burst forward before eventually being denied by a standout save from Gianluigi Buffon.
The hosts should have doubled their lead on the 30-minute mark when Marco Verratti flashed an incisive ball across the box, where Pelle rose highest to send a header a matter of inches over the crossbar.
De Rossi might have secured a brace before the interval, but he headed over poorly after Verratti had again done well to cross from the flank, leaving Italy to go into half time with a 1-0 lead.
The game then descended into chaos in the 55th minute, when De Rossi was shown a straight red for kicking out at Mitsanski, who was also dismissed under somewhat bemusing circumstances.
The pace seemed to be drained from the contest after the dismissals, although Italy did come close to finding a second through El Shaarawy, but he was once again wasteful as he found Mitrev unmoved in his goal.
Bulgaria might have snatched an important equaliser late on, only for Ivaylo Chochev to fire his volley off target with the goal gaping in front of him.