A judge at the High Court in Edinburgh ordered Lukasz Czapla, 41, who killed two-year-old Julius after splitting up with the boy's mother, to serve at least 23 years before applying for parole.
Czapla had denied murdering the toddler in his home in the city’s Muirhouse, in November 2020.
Fuelled by jealousy over his former partner’s new boyfriend, Czapla shot Julius in the head with a BB gun, stabbed him with a skewer-like instrument and smothered him with a pillow.
Following the verdict, Patrycja Szczesniak, Julius' Mother, released at satement through Police Scotland saying that little Julius had trusted his father.
She said: "I only wanted the best for you. If only I could only turn back time to have you in my arms again my child.
"Despite the breakup with Juliusz's dad, I wanted him to have contact with our son. I never stopped him seeing Juliusz. Julek loved him like every child loves their dad. He trusted him, he trusted him completely.
"Juliusz was murdered only for revenge. His dad wanted me to suffer, he succeeded and achieved his goal, at the cost of my beloved little boy’s life.
"If only I knew what was going to happen. I would have never thought he could do this to my child.
"He was only two and a half.
"Every day I think about Juliusz. I can't imagine what his little, helpless heart felt that night and I feel his fear and pain every day. He was part of me, part of my heart and soul.
"I wish I could hug him, play with him and watch him grow.
"Every day I ask myself why there is so much evil in this world, why the innocent have to suffer.
"He not only took Juliusz life but my life too.
"That day, the world stopped, not only for me but also for my mother and my sister.”
Ms Szczesniak continued: “Juliusz was a very happy child. He lit up the place wherever he showed up. He could put a smile on anyone's face and always looked forward to calls with his Grandma and Auntie. They managed to build a strong relationship despite the distance in miles between us and enjoyed every minute they had together.
"It is breaking our hearts that we are no longer able to take his hand and lead him through life. To watch him enjoy riding his bike and eating his favourite ice cream. Listen to his giggling and laughing and being amazed by the things he learned every day.
"There was so much we wanted to show him but this chance was brutally taken from us and now the only thing we can do is to prey for the highest possible punishment. We want to believe in the justice system and that justice will be served, although I know there is no sentence that will bring my angel back to life.
"Czapla should not have the right to any defence and we will fight to ensure that monster never leaves prison.
"My suffering and my pain will stay with me for the rest of my life. But the unconditional love I have for my baby; my dearest child Juliusz will never end.
"My family, friends and l will miss him dearly and we will suffer his loss until the end of our days.
"Juliuszek left us the most beautiful memories that will remain in our hearts forever."
Czapla admitted causing the tot’s death but denied murdering him and a guilty plea to the lesser charge of culpable homicide was rejected by the Crown.
A jury heard Julius was killed by Czapla "in a fit of anger and spite" to "get back at and hurt his former partner" Patrycja Szczesniak.
Prosecutor Alan Cameron said the evidence before the jury showed Czapla was motivated by anger and jealousy over Ms Szczesniak's relationship with her new partner.
Outside court, Detective Inspector Bob Williamson, of Police Scotland’s Major Investigation Team, said: "Julius was a bright, happy young boy and his loss in such horrendous circumstances has been devastating for his mother and his family. Lukasz Czapla is now facing the consequences of his actions. The outcome of this case will never bring Julius back but it does bring this aspect to a close. Julius’s family have understandably asked for privacy during this difficult time.
"I would like to thank the local community and our dedicated team of officers, along with our partner agencies, who assisted throughout the duration of the investigation."