Madni Ahmed, 20, stabbed the grandson 15 times while he was sleeping before stabbing Riasat Bi at their family home in July last year.
West Midlands Police said Mrs Bi was left with more than 25 stab wounds in the “horrendous and violent attack”.
Ahmed, who is from Stoke-on-Trent, was jailed for life at Birmingham Crown Court on Friday 11 January.
He was convicted on Thursday 10 January of Mrs Bi’s murder and the attempted murder of her 18-year-old grandson, who was left with life-changing injuries.
West Midlands Police said Ahmed said he was looking for someone at the house and when he could not find his supposed target, the grandson was attacked.
After the attacks, Ahmed was confronted by another grandson who told him to leave before grabbing a knife to defend himself.
Ahmed refused to leave the house and following a brief scuffle he received a slash wound to his arm.
He then escaped from the house in Aubrey Road, Small Heath, through the attic window, leaving his knife in the roof guttering and jumping into a neighbour’s garden.
Ahmed’s blood was found on the fence panels as he stumbled through the garden and into a neighbour’s house, grabbing a knife from their kitchen.
He was arrested at a hospital in Stoke-on-Trent after detectives traced the mobile phone he had been using.
Detective Inspector Harry Harrison described the incident as “a horrifying attack on a defenceless elderly lady”.
He added: “This was a pre-meditated attack, with Ahmed intent on killing someone that day by taking a knife to the home of an elderly lady.”