Officers remain at the scene of the explosion
The blast happened outside Runnymede St Edward's in Sandfield Park, West Derby - an independent Roman Catholic school.
Witnesses said the device was a nail bomb, but police described it as an improvised firework-type device.
Supt Ian Pilling described the incident as a "despicable" act. He said children had been leaving at the time and it was lucky no-one was killed or injured.
Merseyside Police said a man was seen running away from the scene after the blast, and they appealed for witnesses.
Officers are looking for two men, one of whom is described as white, aged in his forties, stocky, about 5ft 10in tall with light coloured hair.
The school was evacuated immediately after the device went off and pupils were moved to another part of the building. No-one was hurt.
I think if they had actually been inside the vehicle they would have been seriously injured or quite possibly killed
Supt Ian Pilling
The interior of the car was extensively damaged and the windows were smashed by the blast, which happened just 10 minutes before most of the pupils were about to begin leaving.
Police confirmed the silver saloon belonged to a teacher and said the timing of the blast could have proved fatal.
Speaking near the cordoned-off entrance to the private school, Supt Ian Pilling said: "Children were in the process of leaving. Nobody was killed or injured but that was down to good luck."
He added: "I think if they had actually been inside the vehicle they would have been seriously injured or quite possibly killed.
"This type of offence is despicable, particularly as it was close to a school at a time when the students were due to leave.
"This is an extremely unusual offence and I would like to reassure the public that it appears to be an isolated incident."
Students 'shaken'
One student at the neighbouring St Edward's College said: "I am appalled and shocked at what has taken place.
"The area where the bomb went off was very close to where the young primary school children leave the school.
"If the bomb had detonated only 20 minutes later this could have been a very different story indeed."
Another college student added: "This incident was very unusual for the area and the school was shook to it foundations by it.
"The roads were shut off and no-one could get in or out of the school area for a considerable amount of time. I am thankful that no-one got hurt."
Fingertip search
Although he confirmed the device was in a teacher's car, Mr Pilling said there was nothing to indicate why the member of Runnymede St Edward's staff was targeted.
Forensic teams remain at the scene carrying out a fingertip search of the scene and the car.
Runnymede St Edward's is a Catholic independent school for boys and girls between the ages of three and 11, and is Choir School to the Metropolitan Cathedral of Liverpool.
It is on the same site as St Edward's College.
The school, a few miles outside the city centre, has annual fees of about £5,000.
Map highlighting location of Liverpool, England.
A nail bomb has exploded under a car, thought to belong to a teacher, at least ten meters from the entrance of St. Edward's College in Liverpool, England at approximately 14.50 BST. There were no injuries, but the car was severely damaged in the blast.

Merseyside Police confirmed that officers were investigating a security alert at the college which is an independent Catholic boys and girls school for children ages 11-18.  They also confirmed that it was "some kind of explosive device" which was involved.

"There was an explosion under a car at the school this afternoon. What caused the explosion and the reasons behind it we are investigating. Nobody has been injured and there has been no collateral damage," said Michael O'Callaghan, an Inspector for the Scotland Yard. Other reports say that the bomb was inside the car.

Police believe that the bomb was timed to detonate at the time children were being let out of school for the day. Almost 2,000 students attend the school, which was evacuated after the blast.