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After taking the first set 6-2, the number two seed could not dominate and took just four of 19 break points.
World number 67 Giorgi levelled but in game three of the decider, Muguruza won six break points and eventually took one, going on to win 6-2 5-7 6-4.
"It was very tough," said Muguruza, who faces Slovakian Jana Cepelova next.
"I like to play tough matches, it puts me more into the tournament and gives me more confidence."
Giorgi - considered by many to be better than her ranking suggests - was left to rue 42 unforced errors, 12 more than the world number two, who despite her lofty ranking, has only twice progressed past the quarter-final stage of a Grand Slam.
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Five-time Wimbledon champion Venus Williams equalled the record number of appearances at a Grand Slam as she survived an early scare.
The 36-year-old saved two set points when 6-5 down to Croatia's Donna Vekic in the first set before taking a tie-break, going on to win 7-6 (7-3) 6-4.
The world number eight now has 71 Grand Slam appearances, equalling American Amy Frazier's record set in 2006.
Williams is in the same half of the draw as Muguruza but now faces world number 115 Maria Sakkari of Greece in round two.
Sakkari took one hour and 22 minutes to see off China's SaiSai Zheng, almost half an hour less than Williams required to progress in one hour 51 minutes.
"The first set there were some hairy moments, down some set points, but I guess that's where experience sets in," said Williams, playing in her 19th Wimbledon.
This was Williams' first match on grass this year and though she did not hit top gear throughout, she showed killer instinct to hold serve under pressure to take a 5-4 second-set lead before pressing her opponent into a decisive break.
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Germany's Sabine Lisicki, who holds the fastest ever women's serve at 131mph, took just 59 minutes to beat American Shelby Rogers, hammering nine aces at a top speed of 122mph.
Lisicki took just 52 seconds to win the opening game - which included three aces - and her reward is a meeting with 14 seed Sam Stosur, who beat Poland's Magda Linette 7-5 6-3.
Lisicki's brilliance on serve was partially matched by American Bethanie Mattek-Sands, who produced more aces than any woman on day one with 11. But nine double faults  - another day one high in the ladies' draw - meant she still lost in three sets to Lucie Safarova.
The 23rd seed and 2008 French Open winner Ana Ivanovic could not cope with the pressure applied by Ekaterina Alexandrova of Russia.
The 21-year-old qualifier was making her Grand Slam debut but defied her world ranking of 228 to hit 19 winners and take four out of five break points over the course of two sets.
Ivanovic, who reached the semi-finals at SW19 in 2007, apportioned some blame to a wrist injury. "It caused me a lot of miss hits," she said.
Ninth seed Madison Keys, winner of the Aegon Classic earlier this month, did not suffer the same upset as Ivanovic, showing good touch in seeing off Laura Siegemund 6-3 6-1 as she looks to improve on a place in the quarter finals last year.
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assistant: French Open champion and 2015 Wimbledon runner-up Garbine Muguruza won a tense contest against Italy's Camila Giorgi to reach round two at SW19.