The Dutch ran a brilliant race but US. anchor Alexis Holmes refused to let Bol escape and was within inches of her before Bol crashed to the track, hitting her head and dropping the baton.
The winning time of 3:08.80 broke the Americans' own world record set in the first running of the event in 2019. Great Britain took the silver in 3:11.06 and the Czech Republic bronze in 3:11.98 - both national records.
An event still finding its feet delivered an unforgettable finale for the impressive crowd on the opening night of action in Budapest.
Gabby Thomas had been expected to join the US team after sitting out the morning heats but the 200-metre specialist was not included as Matthew Boling came in for Ryan Willie who ran in the morning heats.
The favourites started well but a brilliant second leg by Lieke Klaver pushed the Dutch, who took silver last year, into the lead.
By the time Bol got the baton for the last leg it looked a foregone conclusion as the 400 hurdles specialist and indoor 400m world record holder has been in stunning form in recent weeks.
Holmes, however, delivered a gutsy effort to stay in her wake and was inching alongside her with the outcome still in doubt when Bol collapsed.
"I don't know what happened, I cramped towards the finish line, I was pushing, pushing, pushing," Bol said.
"I was disappointed that my body did not have it in to finish the race strongly. I'm sorry for my team, I should have finished it off and it sucks big time."
For the US, the victory and world record was welcome validation of their one-lap strength in depth after they managed only bronze in the event in Tokyo.
"It is amazing," said Holmes. "It was definitely tough to run against Femke Bol. She is an amazing athlete but I believed I could run it down because I felt really strong and determined to win and I felt good."
Heartbreak for Hassan
That was the second devastating disappointment of the day for the Netherlands, with Sifan Hassan falling while leading within sight of the 10,000 metres finishing line 20 minutes earlier.
Gudaf Tsegay, the reigning world 5,000 metres champion, pulled up tightly beside Hassan down the home stretch and the two ran shoulder-to-shoulder before the Dutchwoman's crash, leaving Tsegay to cross alone in a time of 31 minutes 27.18 seconds to lead an Ethiopian medal sweep.
"When Sifan tried to cross my line not far from the finish, I even did not pay attention to that," said Tsegay, who finished with a stream of blood running from her knee down her shin.
"It happens often at different competitions because every one of us always fights for the leading position. I just had in my mind how to find the way to finish first and, probably, to make a historical finish for our country and to make the Ethiopian people proud of us."
World record-holder Letesenbet Gidey was second in 31:28.16, while Ejgayehu Taye won the bronze (31:28.31).
The 26-year-old Tsegay ran 59.01 seconds in the last lap that saw Hassan blaze by six runners down the back stretch and into the lead - until disaster struck.
The crowd at National Athletics Centre gasped when the Dutch runner wobbled and then went down.
"I think I got pushed by the Ethiopian," Hassan said. "I hope I am not crazy. I will need to look at the replay to see what exactly happened."
She eventually got back on her feet and finished in 11th place.
"I am keeping my smile but it is really hard. I am very disappointed," she said. "This is sport, these things happen. I just had a bad moment.
"I felt really strong and tried to push in the last lap. I really enjoy what I do. I do not think I am injured. My hand is OK but the knee needs to be checked. Hopefully, all is good."
Hassan, who won double gold in the 5,000 and 10,000 and bronze in the 1,500 at the 2020 Tokyo Olympics, is repeating her lung-busting Olympic triple in Hungary, saying on Instagram earlier in the week: "I want to see if I can do it again, because the challenge of running three distances fuels me."
Less than seven hours earlier on Saturday, Hassan won her heat in the 1,500 metres. Her six races, including heats and finals, stretch over eight days.
The Dutchwoman also won the London Marathon in April in her debut at the distance, despite stopping twice to stretch a tight leg muscle.
Tsegay will also run the 5,000, beginning with the heats on Wednesday.
"It was good race with an amazing finish; 10,000 is not my usual event," Tsegay said.
"I like 1,500 and 5,000 much more, but now I feel very special as I managed to add one more world title at a different distance. But 10,000 is not an event where I have a lot of experience. That is why it was not easy to win tonight.
"In any case, I was ready for a very fast finish, because I was not focused on time today, but only on gold medal."