Vincent Uzomah, who is 50 years old and understood to be supply teacher, was attacked at Dixons Kings Academy at 08:55 BST.
He was stabbed once in the stomach with a knife during a science lesson but is in a stable condition, police said.
The teenager was arrested in the city centre following an extensive manhunt.
Det Supt Simon Atkinson, from West Yorkshire Police, said the teacher was attacked in front of other students.
"There were a number of pupils who witnessed this incident, which took place in a classroom, and they are currently liaising with specially trained officers who are obviously supporting them at this difficult time.
"This is believed to be an isolated incident and the police are continuing inquiries to establish the facts," he said.
It is thought Mr Uzomah had only been working at the school for four weeks.
Police said his injuries are not life threatening and his family are at his bedside.
Executive principal of the Dixons Academy group, Nick Weller, said: "It happened right at the beginning of the school day and there were some students who witnessed it.
"Those students have been interviewed by the police. Obviously, it's a very shocking thing for them to see.
"The general atmosphere in the school is very calm, quite orderly, we're trying to keep to normal routines as much as possible."
Mr Weller said there was nothing in school that suggested the attack was about to happen.
He added: "The police are looking into the lead up to this incident, there was nothing in school that led up to it.
"There may have been other people outside the school who maybe knew something."
He confirmed the suspect has been a pupil at the school for a year.
The school, formerly called the King's Science Academy, opened in 2011 and was one of the first free schools to open.
It has about 700 pupils and has since become part of the Dixons academy group.
Education Secretary Nicky Morgan said violence of any kind in schools is "totally unacceptable", adding that her department is "ready and prepared to offer all necessary support".
Ian Murch, of the National Union of Teachers' Bradford branch, which has members at the school, said: "Obviously people are shocked - they would be when an incident like this happens at a school. We hope it can be resolved without any more suffering or difficulties for anybody."
He said the incident "does raise questions with people about safety and what can be done to improve safety".

Mitchell won the women's 5,000m title as she clocked 16:27.08 to finish over eight seconds ahead of Letterkenny's Annmarie McGlynn.
Queen's athlete Mitchell has achieved Commonwealth Games consideration standards in the 5,000m and 10,000m.
Derry Track Club's Kirk-Smith won the men's 3,000m steeplechase in 8:55.72.
Kirk-Smith has also bettered the steeplechase Commonwealth Games consideration standard.
His winning time left him over 22 seconds ahead of runner-up Clonliffe's Jayme Rossiter.
Ciara Mageean eased into Sunday's 800m final with a dominant heat performance.
World Championships-bound Mageean clocked 2:11.92 which left her ahead of City of Lisburn's Kelly Neely, who ran 2:13.13 to also progress.
Letterkenny man Mark English won his 800m heat in 1:53.35 to qualify for the men's decider.
Raheny's Kevin Dooney won the men's 10,000m title in 29:30.16 as he finished over five seconds ahead of marathon specialist Mick Clohisey.
Cork athlete Phil Healy won the women's 200m title in 23.56 seconds while Carrick-on-Shannon's Eanna Madden was just outside 21 seconds as he took the men's title in 21.07 with Clonliffe's Jeremy Phillips second in 21.14.
Marcus Lawler was an absentee from the men's event because of injury with Amy Foster skipping the women's 200m to concentrate on Sunday's 100m.
In-form Brian Gregan led the qualifiers for Sunday's 400m final where Ballymena & Antrim's Craig Newell will also be in action.
Sunday will be the main day of finals action at the Santry meeting with Thomas Barr and Kerry O'Flaherty, Ben Reynolds and Adam McMullan among the athletes who will compete, while Mitchell will aim to complete a double in the women's 1500m final.
A 14-year-old boy has been arrested on suspicion of attempted murder after a teacher was stabbed at a Bradford school.
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Commonwealth Games hopefuls Emma Mitchell and Adam Kirk-Smith both clinched gold medals on the opening day of the Irish Championships at Santry.