Unemployed Amos Price, 30, from Pelsall in the West Midlands, has also been banned from keeping animals for life after admitting animal cruelty.
Magistrates said they would have jailed him for longer had they been able to.
CCTV shows Price, who claimed he was drunk, approaching a house with his lurcher-type dog at about 10.45pm on 23 October.
Price, who was wearing a cap and a hooded jacket and carrying a high-powered torch, pulled the dog towards the cat, triggering the vicious mauling.
The father-of-two’s actions caused outrage on social media when the footage was released to try to identify him.
Cleo the cat’s owner, Gary Truefitt, had viewed the footage after being unable to find his pet.
Mr Truefitt said the attack had affected his sleep and made him not want to go home “because I know Cleo is not there”.
He added: “Cleo would always be there in the morning but now she is not. I feel as if a big part of my life is missing.”
Price, who was jailed last year for criminal damage and assault, cooperated with police when they arrested him on Tuesday, saying: “I was going to hand myself in on Friday. I am done with looking over my shoulder and my family are getting threats.”
Mandeep Sehmi, defending, said Price would not have committed the offence if he had been sober and was extremely ashamed and sorry.
“His family and he have received death threats,” Mr Sehmi told Walsall Magistrates’ Court.
“He knows that his being in custody, his family will have to face such threats alone.”
Chairman of the bench, Brian Benton, told Price that while the court was restricted to a maximum sentence of 26 weeks, it would have imposed a longer period had it been able to.
“Due to your guilty plea, you are entitled to a reduction of one third, to 18 weeks,” Mr Benton said.
“However, due to the circumstances, we would if we were actually permitted to do so have imposed a far greater custodial sentence.”
In a statement released last week, Mr Truefitt said: “Cleo will be very much missed.
“She was a daft cat in lots of ways and would lie along the driveway even when there were cars about, but she was a companion to me.
“It’s just awful to think of her being attacked in this way.”