Five-time Paralympic champion Cockroft won the women's T34 400 metres for her third title at the 2017 championships while Hermitage triumphed in the T37 400m in a world record time of one minute and 0.29 seconds.
Cockroft has already won the T34 100m - in a world record time - and T34 800m and has never lost a race at a major championships but the 24-year-old admitted she battled illness to win.
Cockroft, who set a new championship record in 58.29 seconds, said: "I'm feeling really emotional. I even had a little cry. I was confident until yesterday when I got full of cold. When I came to the start line I felt like I had nothing to put into the race.
"Ten just feels like a number at the moment. Ten world titles is a pretty good thing to get. I'm not 100 per cent, I'm full of cold and I'd been in bed all day just trying to get enough energy together.
"All the way round I was thinking 'I can hear the girls coming. I'm not going fast enough, I'm not going fast enough.' There's no guarantee I'm going to win the gold any more but people seem to think I will.
"I'm not finished yet. My motivation was out there, those two young girls right at my back. They are the reason I train.
"I want to prove I can still go fast. I just want to keep going out there. I want to prove I can keep going a little while longer."
Hermitage defended her world title after winning in Doha two years ago and added to her T37 100m Rio gold from last year's Paralympics.
"It's what I came for and very relieved it all worked out. To seal the deal in the 400 is beyond my expectations," said Hermitage, who runs in the 100m on Saturday.
"I'm competing against girls who are 17, I'm nearly old enough to be their mother so I wanted to prove a point there's life in the old dog yet.
"I'm well capable (of going below a minute) but on the big stage it's just not there.
"That feels like a really long straight out there and I was totally spent by the end. I wasn't sure at all about the time so was very surprised to see it was a world record.
"It's so unbelievable, it's been such a difficult time for me this season so I just decided to go out there and give it my best shot."
Great Britain's Kare Adenegan,16, also claimed bronze behind Cockroft in the T34 400m.
"This is a huge experience for me," she said. "I've learned so much about tactics, I just need to work on my strength and come back fitter and better. I can go home now and enjoy the school holidays and have a nice break."
Kyron Duke picked up silver in the F41 shot put while Stephen Miller won bronze in the F32 club throw, Richard Chiassaro took bronze in the men's T45 400m as did Isaac Towers in the T34 800m.
