Flight attendants bumped their heads and luggage spilled out of overhead binds during the incident Sunday evening.
The Airbus 319 jetliner took off from Detroit Metropolitan Airport with 126 passengers and a crew of five, bound for Dallas-Fort Worth International Airport. It was over Tecumseh in southeastern Michigan about 8:22 p.m. when controllers reported another plane nearby, the FAA said.
"Air traffic controllers notified the Spirit pilot that a skydiving jump plane was climbing just south of the jetliner's position," FAA spokeswoman Elizabeth Cory said in an email. "The Spirit pilot confirmed that he could see the smaller aircraft on his Terminal Collision Avoidance System. ... A minute later, the Spirit jet received an automated TCAS warning that required him to begin an immediate 1,600-foot descent to 12,800 feet from a previous altitude of 14,400 feet."
At the closest, the two planes were 1.6 miles apart horizontally and 400 feet vertically, Cory said.
The sudden dive caught the passengers unaware, said Janet Dunnabeck of Whitney, Texas, who was returning with her 10- and 19-year-old daughters from a visit with Michigan relatives.
"It was horrifying," Dunnabeck told The Associated Press. "Every person on that plane was screaming. We thought we were going down."
Dunnabeck said the plunge caused overhead luggage bins to spill open, drinks to spill and flight attendants to bump their heads.
Two of them complained of pain and asked for medicine, sitting down while nonworking flight attendants stepped up to help.
Jolene Dunnabeck, 10, said she was "really scared."
"It felt like we were falling, we were going to hit the ground and die," she said.
Only after the dive was the pilot able to give out information, announcing only that a "flight control issue" led to the maneuver.
"Thank God he was able to control the plane," said Janet Dunnabeck, who said she spoke with the pilot later at Dallas-Fort Worth airport.
No passengers were injured, said Spirit spokeswoman Misty Pinson.
"Our pilots followed appropriate procedures and adjusted their flight path upon receiving an advisory of another possible aircraft in range," Pinson said in an email. "The flight continued to Dallas/Fort Worth without incident."
Addressing why the two planes got dangerously close, the FAA pointed to the smaller plane's pilot.
"The skydiving plane was flying under Visual Flight Rules, under which pilots are responsible for seeing and avoiding other aircraft," Cory said.
Gabrielle Dunnabeck, 19, said she, her mother and sister want answers about what went wrong and whether air traffic controllers should have alerted the Spirit's crew sooner.
"We're still wondering who dropped the ball," she said.
Broadcast editor Jennifer Garske in Washington assisted with this story.