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The British pair squeezed past their opponents 2-6 7-6(5) 11-9 and into the second round.
Murray, widely regarded as one of Britain's all-time great sportsmen, won Olympic singles gold in 2012 and 2016, apart from capturing three Grand Slam titles in a glittering career.
The Tunisian won 6-1, 6-2 to storm into the third round.
Murray held his nerve in the deciding 10-point tie-break, closing out the match with incredible fortune as his bunted forehand return caught the net and left Berrettini no chance.
Three-time major winner Murray was set to face Djokovic for the 37th time in a highly anticipated clash on clay at the ATP Masters 1000 event in the Spanish capital.
The tennis maverick, Kyrgios, returned to the combustible John Cain Arena and gave raucous fans a night to remember.
The theft prompted the Scot to issue a video plea on Twitter in an attempt to recover the ring.
Murray, who married Kim Sears in 2015, said in a video posted on Instagram that the ring had been returned after he made calls to hotel security.
London: Former championsVenusWilliams and Andy Murray were named in Wimbledon's initial list of wildcards on Wednesday for this month's grasscourt Grand Slam. American Venus, who will turn 41 on Thursday, has won five out of her seven Grand Slam singles titles at Wimbledon but has slipped to 103rd
Murray decided to withdraw from next month's Australian Open following a positive test for COVID-19 earlier this month.
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