Paul Rich assaulted three-month-old Patrick Bradley during a "spur of the moment" attack after becoming "frustrated" because he would not stop crying.
The 53-year-old had tried claiming Patrick's fatal head injuries were caused when he dropped him from his arms into his Moses basket but later accepted he had unlawfully killed him.
Bristol Crown Court heard the fatal injuries were inflicted in the early hours of March 25 2017 when the defendant was looking after the baby at the child's home in Swindon.
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Patrick's mother and father were out that evening and at 3.15am - shortly after the infant suffered the fatal injuries - Rich dialled 999.
He told the operator: "It's the baby. Just got him out of his cot. He's not looking right to me. His eyes are semi-open and he seems all floppy to me."
Patrick was taken to the Great Western Hospital in Swindon and later transferred to the specialist intensive care unit at Bristol Children's Hospital.
He never recovered from his injuries and his life support was withdrawn on March 30. He died in his mother's arms a short time later.
The cause of death was given as "severe head injury".
Jailing him, Mr Justice Soole said: "Your account of that night has taken a very long time to emerge and the full picture is not complete in every detail.
"Patrick was in a Moses basket on the floor, crying and intermittently screaming. You could not get him to settle.
"You were tired after a full day at work and then being up through the night; and you were frustrated that Patrick would not settle.
"At 3.15am you called the emergency services. The lead paramedic arrived and found Patrick lying on the sofa, unresponsive, pale and blue. It was obvious Patrick was in cardiac arrest.
"The cause of this, as you finally acknowledged last week, was that you had suddenly assaulted him by compression of the rib cage, gripping and squeezing him from side to side; shaking him and throwing him into his Moses basket.
"You have not given any further detail about the immediate events but, as the prosecution accepts, this assault was a sudden outburst of frustration with a crying child and must have been over within a very short time, perhaps a matter of seconds."
The judge added: "From all that I have read and heard about you, this terrible event was utterly out of character.
"I have read and taken account of the moving victim personal statement. I am conscious that no sentence of the court can ease their deep grief and anger."
Rich, of Sandy Lane, Shrivenham, near Swindon, had been due to stand trial accused of murder and two charges of causing grievous bodily harm with intent on the infant between March 23 and 26 2017.
But following his guilty plea to manslaughter, the prosecution accepted the not guilty plea to murder and the two other charges were ordered to lie on file.
Adam Vaitilingam QC, defending, said that by the time of the fatal attack, his client had been without sleep for 19 hours.
"We are dealing with a few seconds of exasperation or frustrated behaviour, which involves him throwing Patrick into the Moses basket," he said.
"The phrase 'end of his tether' would be a fair description in this case."