Laura Cunliffe, 23, admitted putting black-and-white Mowgli into the oven and turning it on for five minutes after finding the remains of the fish on the floor, Barnsley Magistrates' Court heard.
Cunliffe said she rescued it after about a minute, when she realised what she had done, but the kitten died later.
District Judge John Foster said he had to jail Cunliffe today as a deterrent - a decision that caused outrage among her family in the public gallery.
Mr Foster had been told by Cunliffe's solicitor, Alan Greaves, that his client has longstanding problems with psychotic depression and has been sectioned under the Mental Health Act 20 times.
As the defendant was led away in handcuffs, one member of her family shouted: "She doesn't know what's happening - she hasn't a clue."
Another shouted: "This is a failure of medical services, judge."
Mr Foster told the defendant: "This was an act of utterly horrendous cruelty on your part on an animal that, as far as I could see, had come to trust you and rely on you."
The district judge said Cunliffe's actions were "inexplicable".
Cunliffe, of Hoyland, Barnsley, admitted one charge of causing unnecessary suffering to an animal at a previous hearing.
She was banned from keeping animals for life and was told she cannot apply to have this ban reviewed for five years.
Brian Orsborn, prosecuting for the RSPCA, told the court that Cunliffe took Mowgli to her stepsister's house after the incident in September last year, wrapped in a towel after attempting to cool the kitten down with water.
Mr Orsborn said the relative described the animal as "proper screeching" before it died and vets said later that it would have suffered a painful death, with terrible damage to its internal organs.
In the days after the incident, the defendant went to Barnsley Hospital, where she also confessed what she had done to nurses on the psychiatric ward.
According to Mr Orsborn, Cunliffe said the kitten had woken her up, because it wanted feeding but, when she went to the kitchen, the goldfish was in pieces.
He said: "She thought that the kitten had killed the fish and she was really upset and angry with the cat and she said she put the cat in the microwave."
The prosecutor said Cunliffe told nurses that she had stopped taking her medication because she did not want to fall into a deep sleep in case the kitten attacked the fish.
But Mr Orsborn told the district judge: "There's nothing to suggest at the time she was actually psychotic."
He said she told herstep sister: "I've put the cat in the microwave."
He said Cunliffe also said: "I looked at the cat and thought 'You evil, f****** bitch'."
Mr Greaves told the court his client was extremely sorry for what she had done and he described it as a "shocking, disturbing and horrendous case".
He added: "She still can't believe what she's done."
The solicitor detailed Cunliffe's extensive mental health problems, which he said were a result of a psychotic form of depression. He said she heard voices and believed that things she saw on TV were real.
Mr Greaves said his client had stopped taking her medication because "the Queen had told her not to".
He said Cunliffe's actions were all the more inexplicable because she loved animals and had cared for the kitten, which was less than a year old.
He said Cunliffe would carry the animal everywhere with her under her jacket and had "loved it more than anything else".
But when he was pressed by the district judge about whether his client "knew what she was doing", Mr Greaves said: "I'm afraid that's right."
He said she "just lost her temper" and acted "totally out of character".
Sentencing Cunliffe, Mr Foster said: "I have concluded, after careful thought, that the utterly barbaric nature of the act that you carried out in relation to this animal does demand a deterrent sentence."
The district judge refused an application for unemployed Cunliffe to make a contribution towards the RSPCA's costs, which he was told ran into hundreds of pounds.
After the case, an RSPCA spokeswoman said: "This was such a sad case and clearly this poor kitten would have suffered terribly over a prolonged period of time.
"Sentencing is a matter for the courts but we hope that this means that other animals will not have to suffer in this way in future."
Cunliffe is not the first woman to be jailed for killing a kitten in a microwave.
Gina Robins, 31, was jailed for 168 days at Torquay Magistrates Court, in Devon, in 2011.
The court heard Robins killed the cat in revenge for its owner reporting her abusive partner to police.