Biles, who led the US women to team gold before claiming the all-around and vault titles, was among several finalists to fall off the unforgiving apparatus.
But the seven-time Olympic gold medallist still has a chance to take her overall tally to eight with a fourth Paris gold in floor exercise later Monday.
Meanwhile, D'Amato, with a score of 14.366, celebrated a first ever women's gymnastics gold for Italy, who also saw Manila Esposito take bronze behind China's Zhou Yuqin.
Just how treacherous the 10cm wide beam can be was clear when the first three starters made big mistakes.
China's world silver medallist Zhou lost her balance and had to touch the beam to avoid coming off.
American medal contender Sunisa Lee and Brazilian Julia Soares both fell, Lee taking a hard fall onto the wooden beam when her foot slipped at the end of what would have been an impressive aerial series.
Italy's Esposito had a big wobble, and Romania's Sabrina Maneca-Voinea fell twice.
D'Amato had a couple of minor wobbles in an otherwise impressive routine to seize first place just before Biles competed.
The US superstar, again greeted rapturously by Bercy Arena fans, made a confident start, but she somehow went off-line on an aerial series and slipped off, drawing a collective gasp from the crowd.
Visibly disappointed, Biles had to wait an agonizingly long time for her score of 13.100 to be posted.
That was nowhere good enough, Zhou earning silver with 14.100 and Esposito bagging bronze with 14.000.
