The world number two come back after going down a break to Yen-Hsun Lu early in the first set.
Murray became the second British player to progress to the second round after Dan Evans saw off Alexandr Dolgopolov earlier in the day.
He walked off court to a standing ovation, throwing his towel into the crowd.
Evans, 26, will face Roger Federer in the next round.
After his match he spoke about elitism in the sport, saying that he had not experienced it.
He said: “I’m not from an elitist background. My dad is an electrician, my mum is a nurse. Tennis people come from a good background.
“It’s that sort of sport. You don’t just need a ball to play tennis. You do need a bit of money to start playing tennis.”
He said that was why the Lawn Tennis Association was giving away around 10,000 lessons to encourage people into the sport.
While the Brits both appeared quite calm throughout their matches, SW19 saw some on-court tantrums.
One umpire was branded “horrible” and another threatened with legal action. Serbian Victor Troicki screamed that official Damiano Torella was “horrible” and the “worst in the world”, after an overrule gave his opponent Spaniard Albert Ramos-Vinolas, a match point.
He lost the match in the next point.
And Frenchman Gilles Simon complained repeatedly about the moisture on Court One throughout his game with Grigor Dimitrov.
He had a heated exchange with referee John Blom about the rain, half-way through the match, which he lost. And later threatened to “sue everyone in the stadium” if he got injured.
Dimitrov is not allowing memories of his previous success at Wimbledon to distract him from the job in hand.
The Bulgarian has spent the last two years searching for the form that took him to the All England Club semi-finals in 2014, a journey that saw him beat Murray en route.
Since then, consistent results have been hard to come by and now 37th in the world, Dimitrov arrived at Wimbledon with few people talking about him. But after a polished win over qualifier Bjorn Fratangelo in the first round and then another over 16th seed Gilles Simon in the second, he is starting to enter the conversation.
“Absolutely - with each match you get more and more confident,” Dimitrov said after his 6-3 7-6 (7/1) 4-6 6-4 win.
“You start to feel the grass more, your movement more, your shots, your routine. Everything becomes very natural. You get in a good rhythm. So that gives you that calmness when you come out on the court.
“But honestly I’m trying not to think so much about what happened two years ago, because if I just keep playing on that - ‘Oh, I had an unbelievable Wimbledon two years ago’ - I will never be able to do better.
“In a way, I just don’t want to live in the past.”
The future for Dimitrov is a third-round meeting with Steve Johnson.
British number one Jo Konta lost her second round match against Eugenie Bouchard 6-3 1-6 6-1. It means Konta is yet to make round three at the All England Club while Bouchard goes through to meet Slovakian Dominika Cibulkova.