At Liverpool Magistrates' Court, Garron Helm, 21, admitted sending an offensive, indecent or obscene message.
He was ordered to pay an Â£80 victim surcharge after the judge upgraded the offence to one considered racially motivated.
Labour MP Ms Berger said the sentence "sent out a clear message".
The Liverpool Wavertree MP said: "Hate crime is not tolerated in our country.
"I hope this case serves as an encouragement to others to report hate crime whenever it rears its ugly head."
Helm, of Thomson Road, Litherland, Merseyside, sent the message in the early hours of the morning of 7 August.
It had been sent from a Twitter an account called "Aethelwulf", which means "Noble Wolf" in Old English, the Crown Prosecution Service (CPS) said.
The tweet was investigated by Merseyside Police's Cyber Intervention Unit and was traced back to Helm.
A search of his home revealed Nazi and far-right paraphernalia, the CPS said.
Helm was arrested and charged with sending an offensive message via a public communication network.
Claire Lindley, from the Crown Prosecution Service, said: "It is vital that the right to freedom of speech is upheld but this tweet was so grossly offensive that it crossed the threshold of criminality and was clearly targeted at Luciana Berger because of her religion.
"This is a religiously-aggravated hate crime and it won't be tolerated."
A man with links to an extremist right-wing group has been jailed for four weeks for sending an anti-Semitic tweet to Liverpool MP Luciana Berger.