Vickilee Galloway, 37, set her dog on Stacey Findlay after the two women started arguing in February last year.
Part of Ms Findlay's ear was torn off and a man was also attacked.
Galloway admitted assault by inciting her pet to attack Ms Findlay. She was jailed for three years at the High Court in Edinburgh.
Judge Lord Kinclaven told her: "There is no alternative to a custodial sentence and it requires to be a significant custodial sentence."
The judge ordered that the dog be destroyed, as he was not satisfied it would not constitute a danger to public safety.
Galloway had also admitted being the owner of a dog that was so dangerously out of control that it attacked Brian Pennie.
Both victims were left with parts of their ears missing and needed hospital treatment.
Advocate depute Gillian Ross had previously told the High Court in Aberdeen that the incident happened after Galloway fell out with Ms Findlay while drinking in Mr Pennie's flat.
She said: "The accused started telling the complainer that her name was not in fact Stacey."
Ms Ross said Galloway then told her pet dog Jake to attack the other woman.
She told the court: "She used the words 'get her', 'Jake attack', or something similar.
"She was bitten on the arm which caused her to fall to the ground.
"The dog then started biting and attacking her left ear."
Galloway eventually managed to get the dog under control by grabbing his collar after it injured Mr Pennie and an ambulance was called.
Defence counsel Emma Toner said Galloway had been a responsible dog owner in the past.
She said: "Her position is she had no expectation that the dog was capable of causing such injuries or would behave in that manner.
"The dog was never trained by her to attack anyone."
Miss Toner argued that a non-custodial disposal was appropriate for Galloway who was considered suitable for a community payback order.
Judge Lord Kinclaven told Galloway, who sat dabbing her eyes throughout the proceedings, that he would also ban her from owning a dog for life.
He said: "By your actions you have indicated you are not a fit and proper person to have custody of a dog."
A dog owner who admitted ordering her Staffordshire bull terrier to attack another woman in Peterhead has been jailed.