Emergency medical technicians met the flight when it landed and treated several passengers. There was no indication any of the injuries were severe.
Southwest said it has removed the aircraft from service.
It comes just months after 30 passengers suffered ear and nose bleeding because crew forgot to switch on the cabin pressure.
The Jet Airways flight was travelling from Mumbai to Jaipur on September 20 but turned back shortly after takeoff.
Videos taken inside the aircraft showed the chaos on the flight as air pressure dropped and oxygen masks came down.
Another passenger Satish Nair tweeted a picture of himself with an oxygen mask on, and complained that "the safety of passengers had been completely ignored" by the airline.
Once he had disembarked the aircraft he said: "[That was] horrifying and the worst incident of my life."
Passenger Joel D'Souza told CNN: "I saw many people having nose bleeds.
"I took off my oxygen mask for one second and felt so stuffy and hot in the cabin. There was no announcement by the crew and nobody knew what to do. It was more like a flight to hell."
The Boeing 737 aircraft, which was carrying 166 passengers, landed safely.
Passengers were immediately attended by doctors and given first aid.
Jet Airways said it had taken the cockpit crew off duty pending an investigation into the incident.
- with AP
