Rodman, 22, struck in stunning fashion just before half-time in extra time as the USA won 1-0 in a tense last-eight tie in Paris.
The result keeps them on course to claim a record-extending fifth Olympic women's football gold medal in their first tournament since English former Chelsea boss Hayes took over in May.
"Emma is just obviously really intelligent and she wants us to thrive the way that we always have, and I think that is something she embeds in the way she coaches," said the right-winger, the daughter of former NBA star Dennis Rodman.
"She doesn't want to change anybody's style. She wants everyone to be creative in their own ways and she lets that happen while also trying to put her structure and her principles in, sprinkled in there.
"Allowing us to play freely has been extremely successful."
Hayes brought a rejuvenated squad to the Olympics after notably leaving out veteran star Alex Morgan.
A team that flopped at last year's World Cup has benefited from the return to fitness of Mallory Swanson, who has formed an exciting attacking trio with Rodman and Sophia Smith.
They have eight goals between them in four games at the Olympics, with Rodman netting her third of the tournament to sink Japan.
"Our team is so great. I keep saying this but we want to just keep improving and that is what we are doing," Rodman said.
Hayes, meanwhile, opted not to single out Rodman, preferring to highlight the patience shown by her new charges against what she called an "exceptional" Japanese outfit.
Japan sat back and attempted to thwart their opponents in what was a repeat of the 2012 London Olympic final which the USA won. Hayes said she fully expected the Japanese to adopt that approach.
"A million percent, and anyone who thought otherwise is naive. They are exceptional. Their block is the best in the world at doing that," said Hayes, whose team now head to Lyon for the semi-finals with the aim of being back in Paris on August 10 for the gold medal match.
"I thought we had the toughest of the quarters. I love Japan. I really rate them and I think that they played a brilliant match and they brought out our best patience which was the most-needed skill today.
"It is not always flashy and it is not always what fans want to see, but this is football and it requires different tactical abilities from game to game. We played the right game for the right opponent."
