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The Tokyo Olympic middleweight champion won 60-54 following six two-minute rounds in the welterweight division.
Former kickboxing world champion and ex-Wales international footballer Price started confidently from the first round.
Gudstensdottir took a while to get going and was reliant on a few big shots, but Price was dominant throughout, using good hand speed to take the bout to the final round and claim victory.
Price paid tribute to her supporters, and especially her grandparents.
“Definitely (it was everything I hoped for),” she told Sky Sports.
“I just wanted to relax while I was in there and not rush my work, get six rounds under my belt and let’s go again.”
She added: “(It was) absolutely amazing. I’d like to thank everyone who supported me, who bought tickets to come and watch me, and I really enjoyed the ring walk as well, my first one, so I’m buzzing.”
Price’s grandmother is due to watch the fight in the morning and when asked for a message, Price said: “Just that I love her and if it wasn’t for her and my grandad then I wouldn’t have achieved anything in my life.”