Wildcard Murray, the 2013 and 2016 champion, played some vintage tennis against Basilashvili but from 5-0 ahead in the third set he lost seven successive games before settling down to win 6-4 6-3 5-7 6-3
Defending champion Novak Djokovic began his quest for a sixth title with victory over British teenager Jack Draper on Monday but suffered an early scare as Wimbledon returned after a two-year absence on Monday.
Andy Murray, playing his first singles match at the All England Club for four years, also had a fright but delighted the home fans with a four-set victory over Nikoloz Basilashvili.
Djokovic, unbeaten at Wimbledon since 2017, was caught cold on a slippery Centre Court as wildcard Draper threatened a huge shock on his Grand Slam debut by taking the first set.
But Djokovic, bidding to equal Roger Federer and Rafa Nadal's record of 20 Grand Slam titles, was unruffled as he moved through the gears to win 4-6 6-1 6-2 6-2.
Wildcard Murray, the 2013 and 2016 champion, played some vintage tennis against Basilashvili but from 5-0 ahead in the third set he lost seven successive games before settling down to win 6-4 6-3 5-7 6-3.
On a rain-hit day, twice women's champion Petra Kvitova suffered a first-round loss to American Sloane Stephens while on the men's side the day's big casualty was French Open runner-up and third seed Stefanos Tsitsipas who was trounced by American Frances Tiafoe. Highlights of day one of the Wimbledon tennis championships on Monday (all times GMT):
MURRAY SURVIVES SCARE TO BEAT BASILASHVILI
Andy Murray lost seven games in a row after appearing on the brink of victory against Georgian Nikoloz Basilashvili but overcame the wobble to win 6-4 6-3 5-7 6-3.
In his first singles match at Wimbledon for four years Briton Murray was cruising when he led 5-0 in the third set before going off the boil.
The twice champion regrouped, however, and got the job done after the Centre Court roof was closed.
SWIATEK PASSES HSIEH TEST
Seventh seed Iga Swiatek of Poland overpowered Taiwan's Hsieh Su-wei 6-4 6-4 to secure her first main-draw win at Wimbledon. Joining her in the second round was Sofia Kenin, who made short work of Wang Xinyu 6-4 6-2.
Russia's Andrey Rublev, the fifth seed, advanced with a 4-6 6-4 6-1 6-2 victory over Federico Delbonis while Spain's Roberto Bautista Agut overcame John Millman 6-2 3-6 6-3 7-6(4).
STEPHENS KNOCKS OUT FORMER CHAMPION KVITOVA
American Sloane Stephens knocked out twice Wimbledon champion and 10th seed Petra Kvitova 6-3 6-4.
TIAFOE SENDS TSITSIPAS PACKING
Unseeded American Frances Tiafoe produced the first upset of the Championships, crushing third seed Stefanos Tsitsipas 6-4 6-4 6-3 in the Greek player's first match since his defeat in the French Open final.
Among the women, 2017 champion Garbine Muguruza sent out an early warning to her rivals with a blistering 6-0 6-1 win over Fiona Ferro, while American 23rd seed Madison Keys beat local hope Katie Swan 6-3 6-4.
DJOKOVIC SHAKES OFF RUST TO DOWN DRAPER
Novak Djokovic's title defence began with the Serb losing the opening set against British wild card Jack Draper, but the world number one roared back to seal a 4-6 6-1 6-2 6-2 win to reach the second round.
STANDING OVATION FOR OXFORD COVID-19 VACCINE DEVELOPER
Sarah Gilbert, who co-designed the Oxford/AstraZeneca COVID-19 vaccine, was given a standing ovation before the first match on Centre Court.
She and her colleagues were among the guests invited to the royal box for the first day of the championships.
Guests also included Hannah Ingram-Moore, daughter of the late centenarian fundraiser Captain Tom Moore, who raised over 32 million pounds ($45 million) for Britain's National Health Service.
MATCHES CANCELLED DUE TO RAIN
Persistent rain on the opening day has forced 16 matches to be cancelled across the outer courts. These matches include five-time champion Venus Williams' opening round encounter with Romania's Mihaela Buzarnescu.
Eighth seed Karolina Pliskova must also wait to get her campaign under way against Slovenia's Tamara Zidansek.
SABALENKA CRUISES PAST NICULESCU UNDER ROOF
Second seed Aryna Sabalenka of Belarus got her campaign off to a blistering start, beating Romania's Monica Niculescu 6-1 6-4 in 75 minutes under the retractable roof on Court One.