Froome had two choices: pour all his energy into catching them or lose his overall race lead and its yellow jersey that has already changed hands three times since the Tour started in Germany on July 1.
"Panic stations," he said. "I really thought that that could be the yellow jersey changing shoulders again."
Like a hound chasing prey, Froome hared off after Romain Bardet, Fabio Aru and Rigoberto Uran — the three riders all within 30 seconds of Froome in the overall standings of the Tour that, after a ho-hum beginning, has become thrillingly close.
Earlier at the Tour, Froome's rivals had waited for the race leader to catch them back up when he suffered another mechanical problem, that one with his gears.
There was no such politeness this time.
Cheered on by partisan crowds on the 8.3-kilometer (5-mile) slog up the steep Col de Peyra Taillade — scaled for the very first time by the Tour — Bardet's French team AG2R put the hammer down.
Further back, Froome realized that if he didn't catch them by the top, he might never do so.
The race was on.
Helped first by teammates Mikel Nieve and then by Mikel Landa, and booed by some spectators as he labored past them, Froome worked furiously on the climb to reel in Bardet's group.
"They all emptied themselves to get me back into the race," Froome said of his teammates. "I had to get back by the top of the climb. Otherwise it was game over for me."
"It was a stressful moment," Froome said. "I thought I might not get back to the front."
Froome said the back-wheel problem seemed to be a broken spoke. "The wheel wasn't straight anymore," he said.
By recovering from the misfortune, Froome now takes the jersey and an 18-second lead over Aru into Monday's rest day, the last of two at the Tour, ahead of a crucial last week of racing in the Alps and with a time trial in Marseille.
The stage itself was won by Bauke Mollema of the Netherlands, with a courageous solo breakaway at the front of the race.
Mollema, a top-10 finisher at the Tours of 2013, 2014 and 2015, sped away on the descent from the Peyra Taillade climb and endured over the last 30 kilometers (20 miles) in front of a group of four riders who laid chase.