Stephen Clarke, 59, of Wednesdbury, admitted causing the death of Farzana Kousar, 39, by dangerous driving.
The High Court in Glasgow heard how his Mercedes Sprinter van drifted across the carriageway and killed her.
Judge Lord Boyd deferred sentence until 30 January at the High Court in Edinburgh and continued Clarke's bail.
The court was told Clarke had travelled more than 200 miles before ploughing into the car which mother-of-four Ms Kousar was in on the hard shoulder of the motorway on 22 December 2014.
She was getting out of the passenger door to check the car because the engine was overheating when she was struck.
She repeatedly said "I'm dying" while lying injured on the road before she lost her life.
The court heard self-employed Clarke was a courier driver with UK Express Logistics, sub-contracted since August that year.
Advocate depute Bruce Erroch told the court he had left Coventry at about 01:20 and stopped at Southwaite services at about 05:00.
Clarke was on his way to Glasgow when the collision happened on the motorway where Ms Kousar was a passenger in a car parked on the hard shoulder.
She had opened the door of the car and was making her way out of the Toyota at about 05:30.
Just before her car stopped, a lorry driver, Frank Keiller, saw Clarke's van overtake him then "drift into the middle lane" forcing him to brake.
Mr Erroch said: "The van drifted back out again and then into the hard shoulder.
"Mr Keiller saw Clarke's van appear to 'jolt' and accelerate during these manoeuvres, causing him to think that the driver was asleep.
"Clarke accepts that he was indeed falling asleep at this point of his journey and that he had fallen asleep at the point of collision."
Clarke's van veered fully into the hard shoulder and struck the Toyota, sending it into a spin before it ended up in the middle lane.
Immediately after the crash, passers-by stopped to help and Craig Usher saw Ms Kousar lying on the road.
The court heard the driver of the car was uninjured and Ms Kousar was able to tell Mr Usher that she had pain in her back and that she was dying.
She lost consciousness and, despite efforts by Mr Usher to save her, paramedics pronounced her dead at 06:11.
Clarke told those nearby: "Oh God, what have I done? Is she dead? I didn't see the car."
He told police he was the driver of the van and said: "I don't think I was on the hard shoulder. I may have been. I hope not."
In March this year, during a police interview, he was asked if he had fallen asleep and said: "I hope not. I don't think I did. I'm not going to say I didn't."
A post-mortem examination revealed Ms Kousar's cause of death was chest injuries due to the collision.
Defence counsel Matthew Jackson said the working practices of the sub-contracting firm were investigated after a similar fatal crash near the same area took place a month earlier.
He said the investigation into that suggested fatigue had been a contributing factor.

23 October 2015 Last updated at 02:35 BST
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