Watson, 22, appeared to be cruising into the third round after building a 5-0 lead in the second set, but Italian Giorgi won five straight games to level the set.
However, the British No1 held firm to wrap up a 7-5 7-5 win and set up a meeting with Agnieszka Radwanska in the last 32.
“It was a very tough match as I knew it would be,” said Watson. “Camila’s a great player and she’s shot up the rankings this past year.
“I didn’t feel I did much wrong [in the second set], she played too good, hitting winners and aces. I just had to stay tough and keep fighting.
“I feel that I’m playing better and better as each match goes on and hopefully I can play even better in my next match.”
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