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The second-round clash was the world number one’s first match since he won his seventh Australian Open title in January.
Djokovic went down an early break to the 128th-ranked Fratangelo, but leveled the set with a break for 5-5. He had to dig down again in the tiebreaker, taking a 4-1 lead before Fratangelo fought back to lead 5-4.
But Fratangelo coughed up a double fault and sent a forehand wide on his own serve and Djokovic was able to seal the set on his first opportunity when Fratangelo sent a forehand wide.
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“To be honest, I was a bit nervous in the beginning, I didn’t play my best. Credit to Bjorn for coming out firing from every corner.
“I was lucky to get the first set. After that things went better for me,” said Djokovic, who dominated the second set to book a third-round meeting with German Philipp Kohlschreiber, a 6-4, 6-4 winner over Australian Nick Kyrgios.