Zakariyya Farah targeted the MP in December 2015, hurling abuse down her answer phone.
The 27-year-old, who lives in Tottenham, was jailed at Southwark Crown Court after being found guilty by a jury of making malicious communications.
An audio recording of the telephone call was seized by police and an investigation was launched.
Farah was arrested at his home in January 2016.
The recording was analysed and compared to the voice of the defendant.
A voice expert confirmed there was evidence to suggest he was responsible for the message and he was charged with making malicious communications in November last year.
During his trial at Southwark Crown Court, Farah gave evidence protesting his innocence.
He denied knowledge of the phone call being made, despite agreeing that the phone used to make the call belonged to him.
But the jury convicted him of a charge of making malicious communications and Judge Martin Griffith sentenced him to 15 months in jail.
The incident came just days after Ms Allan had voted with her Conservative colleagues in favour of a bombing campaign by the UK military in Syria.
Ms Allan had publicly complained that she was enduring abuse from constituents about her vote in favour of military action, posting examples of the messages she had received.
But she was accused of doctoring a constituent’s email on her Facebook page.
She became the centre of a social media row after it was claimed that she had added the words "unless you die" to the end of a message she posted.
Detective Constable Karl Latham, who led the investigation for Haringey CID, said: “Malicious communications will not be tolerated.
"This investigation goes to show that these matters will be investigated in their entirety and the offenders brought to justice and put before the courts to face justice.
"I hope that this case serves as a warning to others who think they can get away with abusing or threatening an MP or anyone else for that matter.”