Olympic and defending champion Mu set the pace from the gun but, having barely run the distance this season, was never able to open a gap.
Moraa stayed on her shoulder throughout and delivered her trademark final 100 metres burst to move past the American to triumph in 1:56.03, leaping and punching the air in celebration as she crossed the line.
Hodgkinson, just as she did in Eugene, tried to find a way through on the inside but though this time she got past Mu, was outsprinted by Moraa and had to settle for a second successive silver in 1:56.34, having also taken silver, behind Mu at the Tokyo Olympics.
Mu, whose participation had been in doubt until a few weeks before the championships as she flirted with the 1,500 metres took bronze in 1:56.61.
"After bronze last year I wanted to improve and I have," Moraa said. "Everyone in the final was so fast I knew I would have to have a fast finish. I came from a long way behind but I managed to do it."
"Another podium, another medal," said Hodgkinson, the European champion. "It's not gold but it's not bronze.
"I did think I would get through on the inside but the line came a bit quicker than I thought.
"I was really up for it, I really believed I was going to win. It was a different order this year, it might be again next year. In Olympic year everyone brings even more of their A game. All I can do is aim for gold again. One of these days I will get the top spot."
Mu said: "I came here to Budapest with no big expectations and now I am really grateful for this wonderful bronze medal.
"I wanted to slow the pace a little bit since I knew it would be a killer finish. Once I was overtaken from that point I just wanted to finish on the podium."