The Scot lost the first set 6-4 to the Cypriot before he stormed back to take the last two 6-1 6-4 and book a place in the quarter finals against Nicolas Almagro from Spain after he has kicked off his mixed doubles campaign alongside Laura Robson on Wednesday when they take on Lucie Hradecka and Radek Stepanek.
Murray watched Bradley Wiggins' massive time trial win before stepping out onto Centre Court, in the open air and not under the roof, and he praised the beauty of the Games – getting motivated by being in the same team as other sports.
“Literally right before we walked on court we saw Bradley Wiggins get the gold medal,” said Murray.
“That's the great thing about the Olympics, you get to watch the other sports. It gives you that extra bit of motivation to go out and try and be part of the medal count and do something for your country.
"It was very different conditions as I obviously played under the roof the first couple of matches – and it was very windy today.
”I struggled early on in the match with my timing. I played better in the second set, but it was very tight all the way to the finish."
Murray is expecting a tough encounter against Almagro and knows he needs to start like he finished to ensure victory.
He said: “It’s a tough match. He's playing very good tennis just now and in the top ten in the world in the singles.
“It's going to be a tough match for me and I need to play like I did in the second and third sets today to win that and I can't afford any more slow starts."
Elsewhere, men's number two seed Novak Djokovic struggled into the quarters with a hard-fought 4-6 7-5 6-1 win over the world number 158, Australia's Lleyton Hewitt.
World number one Roger Federer also faced a shaky moment against Uzbekistan's Denis Istomin, saving three break points at 5-5 in the first set before securing his spot in the final eight with a 7-5 6-3 win.
In the women's, Serena Williams ploughed to a 6-1 6-0 win over Russia's Vera Zvonareva, who left the court in tears after the 51 minute annihilation, soon to be joined by team-mate Nadia Petrova after her 7-6 6-4 loss to Belarusian top seed Victoria Azarenka.
Returning to Centre Court after a half-hour rain delay, Beijing bronze medallist Djokovic gifted Hewitt a break at 4-4 in the first set after ploughing a backhand into the net at the end of a long baseline rally.
Hewitt, who won Wimbledon in 2002 but has since been plagued by injury, quickly wrapped up the set to take the lead against the Serbian five-times grand slam winner.
"Incredible match. He started off really well. I don't think I have done anything wrong in the first set ... It was very close and could have gone either way," Djokovic said.
A rattled Djokovic went 5-3 ahead in the second but, serving for the set, he was outsmarted by the 31-year-old, who planted two consecutive backhands down the line to claw back a break as the crowd erupted with a roar.
As cries of "Aussie, Aussie, Aussie, oi oi oi" rang out, the valiant Australian levelled the set at 5-5 before he was broken again, forcing a decider.
Djokovic, who at one point entertained the lively crowd with a shot through the legs, found his stride in the third, breaking the 31-year-old twice to serve for the match at 5-1.
As Hewitt, who was handed his place at the games by the International Tennis Federation, hit a backhand into the net to hand Djokovic the victory, the 25-year-old pumped his fists in the air before letting out a very big sigh of relief.
"It's never that easy against Lleyton, he's one of the best returners ever to play the game. It was a great test, it's always a pleasure to play against him and I look forward to the next challenge," Djokovic said.
That challenge will be Jo-Wilfried Tsonga, who he will play for a place in the semi-finals after the Frenchman mustered enough energy to beat Spain's Feliciano Lopez 7-6 6-4 less than 24 hours after his second round win over Milos Raonic set the record for longest three-set match in Olympic tennis history.
Venus Williams' hopes of becoming the first tennis player to win four Olympic gold medals will have to rest on the doubles after she was knocked out of the singles in a 7-6 7-6 third-round defeat to Germany's Angelique Kerber.
Beligum's Kim Clijsters, who plans to retire for the second time after the U.S. Open in September, moved one step closer to winning a medal at her first and last Olympics with a 6-3 6-4 win over 11th seeded Serbian Ana Ivanovic.
"I feel like I've been playing some really good tennis in the first three matches here, it's just a matter of trying to keep that up and before every match I need to refocus and stay on target," 29-year-old Clijsters said.
"When things go tough, fight each shot and that's all you can do when things are tougher; keep fighting and give it your all."