The 27-year-old was pushed to the brink by world number 69 Watson but outlasted her opponent to set up a quarter-final meeting with Slovenia's Dalila Jakupovic.
"It was not easy for either of us and the conditions made it harder here," Konta said.
"It is never easy playing a compatriot, I feel fortunate because there was very little in it."
World number 156 Katie Boutler, a British wildcard, also surprised the home crowd with a hard-fought 7-6(6) 6-1 victory over Australia Samantha Stosur, who is positioned 53 places above the 21-year-old.
Boutler's reward is a quarter-final meeting with top seeded Ashleigh Barty, who brushed aside China's Yingying Duan 6-3 6-2.
"I feel like I always play well in Nottingham," Barty said. "So hopefully it’s third time lucky, and I can get one step farther in the tournament."
The Australian has been beaten in the quarter-finals in the last two editions of the tournament.
Defending champion Donna Vekic recorded a 6-2 7-6(6) win over Belarusian Vera Lapko and will next take on Germany's Mona Barthel.
(Reporting by Aditi Prakash in Bengaluru, editing by Pritha Sarkar)
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