The fight broke out just after 9pm on Monday outside Asda in Heath Town, Wolverhampton, and left a man in his 40s seriously injured.
He was taken to hospital from the Wolverhampton Road car park but died a short time later.
A 32-year-old suspect, who had left the fight scene, was later found by officers on Old Heath Road and arrested on suspicion of murder.
He had a stab wound to his arm and was taken to hospital for treatment but has since been discharged and taken into custody.
West Midlands Police officers have been carrying out a forensic analysis of the scene, examining CCTV and have been speaking to witnesses.
Detective Inspector Adam Jobson, from the force's homicide team, said: “This was a very sad turn of events and my thoughts are very much with the family and friends of the man who has died.
“Our officers have worked through the night and this morning to seize CCTV, speak to witnesses and gather intelligence. This led to an arrest on a street nearby.
“If there is anyone we have not yet spoken to who witnessed the disorder or the sequence of events leading up to it, we would urge them to get in touch.
“An incident like this can be unsettling for the local community, and I would like to reassure them that uniform officers will be patrolling the area and the public can speak to them about any concerns they may have.”
Asda was open again on Tuesday morning but the car park remained sealed off as specialist police officers scoured the area and forensic officers examined the scene.
A blue forensic tent was put up in the car park on Monday evening, while police tape surrounded the car park. Officers were also seen speaking to staff inside the supermarket.
Meanwhile nearby workers expressed their shock at the man's death but said there was regularly trouble in the area.
A spokesman for West Midlands Ambulance Service said: "We were called to reports of a stabbing on Wolverhampton Road at 9.16pm.
"One ambulance, two paramedic officers and a MERIT trauma doctor attended the scene.
"On arrival we discovered one patient, a man, who was in a critical condition.
"He received specialist trauma care at the scene before being taken to hospital on blue lights and sirens."
Anyone with information has been asked to contact the force via their website, or by calling 101 and quoting log number 4027 of September 6.