Madden, a member of the U.S. teams that won gold medals in the 2008 and 2004 Games, lost her chance at individual gold when horse Via Volo uncharacteristically stopped twice at a fence.
"It was all going well and she was jumping amazing. Then she started fighting me quite a bit and all our good work went wrong," Madden told reporters.
"Luckily, the rest of my team went well, so I must get things right tomorrow."
Horses incur four faults for a refusal and four for knocking down a fence. A pair of refusals means elimination from the individual, but she will still jump in Sunday's team final.
Saturday's round was a qualifier for the individual final on Aug. 8 but riders will not carry their scores forward to either that or to the two-day team final on Sunday and Monday.
The team final consists of two rounds, with the 10 best-performing four-member teams in the first carrying on into the second. The two rounds together determine the winner, with teams dropping their worst score from the two-round total.
Reed Kessler, who at just 18 is the youngest-ever U.S. equestrian team member and Olympic rider, was cool under pressure and jumped clean on horse Cylana. She incurred one fault for taking longer than the allowed time.
Team mate McLain Ward produced an effortless clear on Antares, as did Rich Fellers, a contender for individual gold after winning the World Cup final this year.
"Funny things happen at championships and the Olympics always produce some surprises. For instance, Beezie was my favourite for individual gold and look what happened to her today," said Belgium's Jos Lansink, who also went clear.
Reigning Olympic individual champion Eric Lamaze of Canada also jumped flawlessly, but didn't see himself with a serious podium shot given the inexperience of horse Derly Chin de Muze.
"To be perfectly honest, I'd have to be a very lucky guy for that to happen," he said. "But it's show jumping. The horse is plenty good - I just feel like eventually I may run out of experience from the horse."
Lamaze suffered a shattering blow last year when Hickstead, his 2008 gold medal horse, collapsed and died in the ring at a World Cup event in Verona. He said Hickstead would certainly have been at these Games had he lived.
Team mate Ian Millar, making a record 10th Olympic appearance, had a rail down for four faults while Jill Henselwood had four jumping faults and one time fault and Olympic debutante Tiffany Foster earned eight faults.
The jumping field was more diverse than that in eventing and dressage. A number of Middle Eastern nations fielded riders, as did many Latin American countries.
While British royals were not in evidence this time, Prince Abdullah Al Saud jumped flawlessly for Saudi Arabia on Davos. The team ended up with just three faults on the day.
"I am very proud of the Saudi team and I wish them all the best of luck," he told reporters. "I also like horse racing and follow all of my father's and grandfather's horses."
Syria's Ahmad Saber Hamcho, whose father faces U.S. and European Union sanctions for acting as a financial manager for the brother of Syrian leader Bashar al-Assad, jumped clear but posted one time fault.
He steered away from politics when asked by reporters afterward about escalating clashes between Syrian forces and rebels in Damascus and Aleppo.
"For sure I feel bad, it's my country," he said of the situation in Syria. "But this gives us and all the Syrians positive energy because of what happened now, because of the clear round and we represent all Syrians."
The London-based 19-year-old said the political situation was not a distraction and that his father had told him just to go and ride for Syria.
"I think it made it easier for me because I had more will to do better in the ring there," he said.
It was a mixed day for the home team as Britain's Peter Charles had two down and two time faults for 10 in total and Scottish number one Scott Brash dropped a rail for four faults.
Veteran Nick Skelton and Big Star jumped clear as did Ben Maher on the black stallion Tripple X.