Judge Anthony Leonard QC imposed a hospital order under the Mental Health Act along with a restriction order for the safety of the public.
He said most people would feel “anger and revulsion” for what he did.
Morgan had admitted at an earlier hearing to possessing material likely to be of use to a terrorist.
He also pleaded guilty to encouraging terrorism by composing a song called White Man and sharing it online. The song called on white people to fight and kill people who were not white.
Sue Hemming from the Crown Prosecution Service said: “Tristan Morgan’s violence was partly driven by his hatred of Jewish people which was clear from the library of extremist literature he had.
“The arson attack at Exeter synagogue meant it couldn’t be used while repairs took place and left many of the congregation fearing for their future safety.
“He was suffering from a psychiatric disorder at the time but remains a danger so it’s right that he gets the treatment he needs at a hospital and the public is kept safe.”
The grade two listed building dating back to 1763 is the third oldest synagogue in Britain and was severely damaged during the attack with the cost of repairs rising to £23,656.
Morgan, who was stood close to the window when he set the building alight, was engulfed in a fireball which knocked him off his feet and burnt his forehead and hair.
Police were alerted to the explosion by a passer-by who saw Morgan walking to his van with a green petrol can before driving off.
Officers were able to track him down because the witness took down his number plate and when they arrived at his home Morgan’s first words were: “That didn't take long.”
The arresting officers noticed that he smelt of petrol and his upper right arm was hot to the touch and Morgan said his hands were burning.
He was also wet having attempted to cool his burns by throwing water over himself.
Morgan told police: “Please tell me that synagogue is burning to the ground, if not, it’s poor preparation”.
He also asked officers “Do you believe in race?” after he was arrested and cautioned.
The court heard expert psychiatric evidence explaining how Morgan had been suffering from psychosis at the time of the attack.
It was also concluded the Morgan, whose internet search history included the opening times for the synagogue, posed a high risk of further violence because of his use of alcohol and illegal drugs.
A police search of his home and van revealed 24 knives, a sword, hunting knife and the axe he used in the arson attack.
His laptops, mobile phone and storage devices contained antisemitic and neo-Nazi material, as well as data showing his hatred for people who were not white.
One document he had downloaded was a 340-page terrorist manual aimed at white supremacists to prepare them for a “race war”.
The document included recommendations for targeting synagogues and advice about arson attacks.
Other documents promoted Holocaust denial, “ethnic cleansing” and supposed Jewish global power.
