Mary McCann, 35, from Derby, did not make sure four-year-old Lilly and Smaller, who was celebrating his 10th birthday, had seatbelts on.
Mary McCann, 35, hit the Scania HGV while driving northbound on the M1 near Milton Keynes at about 11.10pm on August 9 last year.
Lilly McCann, four, and Smaller Peter McCann, who was 10 that day, died at the scene, between junctions 14 and 15.
A third passenger, a two-year-old girl, was not badly hurt, while the lorry driver, Simon Denton, was left with whiplash.
She is here to be punished by this court and rightly so, but it must be acknowledged that the true punishment for this offence lives within Mary McCann. It is to that unquenchable wheel of fire that Mary McCann is bound
Laban Leake, defending
McCann appeared at Aylesbury Crown Court via video-link from HMP Bronzefield and wept throughout the hearing.
Stephen Shay, prosecuting, said: “Ms McCann was driving three children in a Vauxhall Astra car when at 11.12pm she collided with the rear of a lorry driven by Simon Denton.
“Tragically, two of her children were killed.”
McCann was in lane one, while the lorry was in lane two, the court was told.
Her car drifted into lane two and, in a bid to avoid the lorry, she crashed into it – sending her car spinning.
Mr Shay told the court McCann was driving at 72mph in a 60 zone, with Lilly and Smaller not wearing their seatbelts.
Lilly was thrown from the car and found on the roadside, while Smaller was flung around the backseat and found slumped in the footwell, the court heard.
A witness heard McCann shout: “God, why didn’t I put their seatbelts on?” in the smash’s immediate aftermath, Mr Shay said.
There is no punishment commensurate with the loss of one’s two children and I accept that. The message must go out: don’t drive under the influence of alcohol
Judge Francis Sheridan
McCann was arrested last September after failing to answer bail following the youngsters’ funerals, the court was told.
Laban Leake, defending, said: “She is here to be punished by this court and rightly so, but it must be acknowledged that the true punishment for this offence lives within Mary McCann.
“It is to that unquenchable wheel of fire that Mary McCann is bound.”
He said his client has complex post-traumatic stress disorder.
Judge Francis Sheridan said Mr Denton did nothing wrong.
He said Lilly would “not have been ejected” if she had been strapped in, while “Smaller would have remained in his seat”.
He said: “There is no punishment commensurate with the loss of one’s two children and I accept that.
“The message must go out: don’t drive under the influence of alcohol.”
McCann, of Bamford Avenue, Derby, was jailed for 49 months and banned from driving for seven years and two weeks after admitting two counts of causing death by careless driving whilst under the influence of drink.