The 35-year-old actress had Norman for 14 years before he died at the weekend.
Posting a black and white photo of herself with Norman on Instagram, Cuoco said she was suffering, “Earth-shattering, deep gut-wrenching pain I didn’t know was possible.”
She also took to her Instagram stories to thank everyone for their well wishes, saying she had “run out of tears”.
Cuoco shot to fame on The Big Bang Theory in 2007, which is around the same she would have first got Norman.
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One of her other rescue dogs, Petunia, died in November last year just months after she and her husband, Karl Cook, adopted her.
Cuoco currently stars in the HBO Max black comedy series The Flight Attendant, which streams on Binge.
She has achieved critical and commercial acclaim in the role, her first big foray into the world of TV since Big Bang wrapped two years ago.
Cuoco told news.com.au in November that she felt pressure to choose the “right” role after more than a decade playing Penny.
“Any project I was going to choose after Big Bang I knew was going to get a lot of attention, good or bad, no matter what I chose people would have thought it was either a good idea or not,” she said.
“I wasn’t necessarily looking for this sort of drama or character but I knew that the right project … I would know deep in my gut.
“So when I saw the book (of the same name) three years ago I simply thought, ‘Wow, this could be a great character for TV.’
“It wasn’t me trying to get away from Big Bang, or move on – it just felt like the right move.
“And the icing on top was I had all these different emotions. I got to play action, adventure, mystery, emotion, but also moments of laughter.”