Their first-round match did not start until after 17:00 BST as downpours delayed the day's play until 15:30.
Watson claimed the only break point of the first set, then saw off six break points for Giorgi in the second.
The world number 55, ranked 11 places higher than Giorgi, will play Barbora Strycova or Karolina Pliskova next.
"I was very focused at the end and I'm so glad we got to play the full match without any rain delays," said 24-year-old Watson.
"She's a very tough opponent and hits the ball extremely hard. I had to accept she will hit a lot of winners but had to take advantage when she made mistakes.
"I feel fully fit and think I'm playing very well."
American seventh seed Madison Keys won 7-6 (7-3) 6-4 against Hungary's Timea Babos in Monday's opening match.
However, British number three Naomi Broady and Australia's Daria Gavrilova were forced off court about 19:00 by more rain.
British number one Johanna Konta and Tara Moore begin their Edgbaston campaigns on Tuesday.
Konta, 25, faces Japan's Misaki Doi, while Moore, 23, plays Austria's Tamira Paszek.
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British number two Heather Watson dug deep to win 6-4 7-5 against Italy's Camila Giorgi at a rain-affected Aegon Classic in Birmingham.