With most of the games being played on Sunday and Monday, Inter knew that a win or a draw at home would lift them, temporarily at least, to the top of Serie A.
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"We did well in both halves. Spezia have played earlier games very well. We knew they would create difficulties for us, but we prepared for the game in the right way," Inzaghi told Sky Sport Italia.
The hosts took the lead in the 35th minute when striker Romelu Lukaku headed the ball to an unmarked Martinez right outside the box before the Argentine forward struck it into the bottom left corner of the net.
Dragowski saved Spezia again in the 40th minute when he stopped a shot from Dumfries with his legs. Only two minutes later, Lukaku headed the ball onto the crossbar.
Inter dominated possession in the first half but only had one goal to show for it, with Spezia not having any shots on target.
Calhanoglu doubled the lead in the 52nd minute after receiving the ball inside the box and calmly slotting it beyond Dragowski.
Correa put the ball into an empty net in the 82nd minute, shortly after coming on, after Edin Dzeko's assist from the left went past Dragowski. The win means Inter top the standings with six points from two games.
Spezia are now without a victory against Inter in all seven of their meetings in Serie A and the Coppa Italia.