Paul John Doherty, 31, a construction worker from Ballynahone in Fahan near Buncrana, was also disqualified from driving for seven years.
He previously had pleaded guilty to committing seven offences during which he knocked down Diane Wilkinson and her daughter Abbie on 21 July last year at Benone Beach.
Derry Crown Court was told that just seconds before the impact Mrs Wilkinson instinctively swept her daughter Abbie up into her arms.
Both mother and daughter were knocked to the sand. Abbie sustained a broken collarbone, a black eye as well as cuts and bruising to her legs.
Mrs Wilkinson was badly bruised and received injuries to her knees and feet. Both Abbie and her mother were held in hospital overnight following the incident.
After he knocked down his victims, Doherty drove his BMW car at speed off the beach. He was chased for eight miles by a police car and during the high chase he forced oncoming drivers to swerve out of his way.
A police helicopter was deployed and Doherty's car was found concealed behind a new build house on the Barnailt Road between Limavady and Claudy.
He'd removed the car number plates and attempted to hide as police officers arrived at the scene.
Following his arrest Doherty made admissions to two charges of dangerous driving, two charges of drink driving and to charges of failing to stop, failing to remain at scene and failing to report an accident.
Judge Philip Babington said Doherty had over twice the legal alcohol limit for driving in his breath.
He also said Doherty had two previous convictions for dangerous driving, both of them in Buncrana, the first in May 2003 and the second on Christmas Day 2005.
He said like the Benone Beach incident, both involved Doherty carrying out high speed handbrake turns in a crowded area.
Describing Doherty's driving on a crowded beach as appalling, dangerous and outrageous, Judge Babington said the Wilkinson family from Newtownstewart in Co Tyrone had been traumatised by the events.
There were many families and young children enjoying themselves and they have every right to think that they would be safe from this type of behaviour in such a location.
"Although Abbie and Mrs Wilkinson were injured in this incident, Mrs Wilkinson's son, six-year-old Joshua witnessed the entire matter," the judge said.
"Mrs Wilkinson has had to have counselling. Her daughter appears to have made a good recovery and generally seems to be making good progress.
"Her son Joshua suffered emotional disturbance for a period of time but again is making good progress."
Judge Babington said Doherty's offending was aggravated by several matters.
"Two persons received physical injuries and consequential psychological upset and one person, who had the misfortune of witnessing this matter suffered psychological upset," he commented.
"There is evidence of alcohol being consumed prior to the driving on the beach and there were a number of passengers aboard the vehicle at all times during the offending.
"The manner of the driving on the beach was particularly aggressive and was obviously contrary to the 10 mph speed limit signs displayed on approaches to the beach and the defendant did all that he could to evade the police."
Meanwhile speaking after the sentencing, Mrs Wilkinson and her husband Richard said they were glad it was now all over for them.
"My daughter Abbie is not too bad but Joshua still suffers from recurring nightmares about it," said Mrs Wilkinson.
"We have gradually tried to re-introduce them to Benone Beach but they are afraid to go to the part of the beach where it all happened. I really don't want to talk about Doherty. He has received a sentence which he deserved."
Mr Wilkinson said the local authority responsible for Benone Beach should introduce tighter speed controls.
He said: "It is a blue flag beach and it must be made safe for families to visit instead of the families having to look out for their children in terms of speeding cars."