A dominant Cockroft, 27, set a new championship record of one minute 57.27 seconds in Dubai.
And 18-year-old Adenegan – who led in the early stages of the race – crossed the line more than four seconds later.
Five-time Paralympic champion Cockroft has now won 12 world titles across all distances.
“I didn’t really have a plan,” she told BBC Sport. “I won it; that was the only tactic behind it.
“Kare really surprised me. She went out strong to the front and she was very welcome to do that because it was quite hot outside.
“I hate being pushed to the front of the race, so she really made the race easy for me. She allowed me to save myself for the last 400m.”
It was Cockroft’s second gold medal of the championships, as she won the 100m title on Sunday.
After that race, Yorkshire’s Cockroft admitted she had “fallen out of love” with the sport in 2018 but on Thursday, she told BBC Sport her fire had been reignited in Dubai.
“This is what I needed – I needed to come here and remind myself of what I love doing and why I love doing it,” she said.
“It’s really done that; it’s motivated me. The girls are still not as far away from me as I’d like them to be, so I need to work hard and go quicker.”
Adenegan – who now has two silver medals in Dubai, having finished second to Cockroft over the sprint distance – beat American Alexa Halko to second place by just three hundredths of a second in a tight finish.
“I am so happy with that,” the three-time Paralympic medallist told BBC Sport. “That was probably the best 800m I’ve had.
“I don’t think I’ve ever beaten Alexa over the 800m because she is really strong. I saw that I was ahead of her but knew I needed to push a bit harder, and it was enough. It was a tough race.”