Lazio continued to see more of the ball in the final third, coming close again on 14 minutes when Michael Ciani hit one from range that almost crept in.
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Hernanes and Cristian Ledesma were next to try their luck in front of goal, but a combination of solid defending from Inter and some poor finishing kept the scores level.
The visitors started slowly but they began to apply some pressure around the half-hour mark. Andrea Ranocchia was gifted a free header inside the box on 32 minutes, only to put it wide.
Inter started the second half brightly and began to take control of the game. Fredy Guarin was particularly lively, causing the Lazio defence all kinds of problems and shooting narrowly wide just after the hour mark.
Lazio's Hernanes almost punished the visitors for their wastefulness on 71 minutes when he forced an excellent save from Juan Pablo Carrizo with a right-footed missile.
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The hosts stole to the lead against the run of play when Stefano Mauri played a deft through-ball to Klose and the German striker fired into the centre of goal for 1-0.
Klose's strike proved decisive as Inter could not find a way through despite having the better of the attacking possession in the second half.