The airline confirmed an “incident” on flight BA422 which departed London Heathrow at 15:10 BST on Monday and landed in Valencia.
Passengers had to slide down emergency chutes to the runway, with one describing the “terrifying” experience as “like a horror film”.
BA has apologised to the 175 passengers on board the aircraft, an Airbus A321.
A statement from the airline said the flight had “experienced a technical issue” as it approached Valencia.
Three passengers were taken to hospital and have since been discharged, BA added. It said staff members had assisted customers in the airport terminal after the evacuation.
BA said there were two pilots and six cabin crew members on the flight.
‘Crying and hyperventilating’
Passenger Gayle Fitzpatrick, who was on holiday with her husband, said: “There were no communications from the crew, some of which started to wear full oxygen masks and protective fire wear.”
“People were crying and hyperventilating. It was genuinely scary,” Mrs Fitzpatrick, from Glasgow, added.
Rachel Jupp, who was on the flight with her children, told BBC News smoke filled the cabin “very quickly” about 10 minutes before its scheduled landing.
Ms Jupp, the editor of BBC Panorama, said there had been no official announcement about what was happening as white smoke appeared to come through the air conditioning system into the cabin.
“Very quickly, you couldn’t see the passengers two seats down from you,” she said.
As the plane began to descend quickly Ms Jupp said she heard calls to “get down” in order to breathe the cleaner air near the floor of the aircraft.
She said the pilot did a “really good job” to make a “pretty smooth landing”.
“We later found out the cockpit was full of smoke and he had a gas mask on,” she added.
Ms Jupp said the cabin crew was unable to open the emergency exits for “three or four minutes” after the plane landed.
Mark Kay, who was travelling with his 13-year-old son, said they felt “trapped and helpless” while staff tried to open the doors.
Passengers then slid down the emergency chutes to the runway.
As members of the emergency services ran towards the plane, Ms Jupp said: “We were just told to run and get as far away as we could from the plane.”
Mrs Fitzpatrick said fire crews were waiting on the runway when the plane landed and passengers were directed to the terminal.
Another passenger at Valencia airport told the BBC a member of staff on the ground had said there had been a “fire in the motor” of the aircraft.
Daniel Kietzmann, who was on the plane with his wife, said the situation was “poorly handled” by staff and there was a lack of communication about what was happening.
“Nobody was on hand when we went down the slide so my wife hit the tarmac very hard,” he said, adding that he and his wife were bruised.
“We were left for hours in the terminal with ground crew that had no idea what was happening and nobody spoke English.”
BA said all passengers were evacuated safely by its crew and met by the airport’s emergency services.
“In addition to our team on site, other British Airways team members have arrived in Valencia to help our customers and our local airport partners with anything they need,” the airline said in a statement.
In an email to affected passengers BA said it was “sourcing an alternative aircraft” to operate flights back to London.
Travellers who were delayed overnight were given free accommodation at a local hotel, BBC News was told.