He followed the driver to Remuera Rd where both men got out of their cars and fought.
"The defendant proceeded to hit the victim in the head with a steel tyre iron, causing a deep laceration to the chin of the victim requiring stitches," a summary of facts said.
Zajmi put the implement in a second vehicle being driven by his wife and it was never found.
He told police at the time he had only punched the man but today pleaded guilty to the police's summary.
His lawyer, Jeremy Bioletti, indicated he would be making an application for a discharge without conviction at sentencing in October.
His client wanted the chance to complete a number of "programmes" before that date, he said.
Bioletti also argued his client should be granted interim name suppression while he was on bail awaiting sentencing.
Zajmi and his wife bought their trendy central cafe in 2013 and another one at NorthWest Shopping Centre eight months ago.
"From a business point of view, he's concerned of the impact of publication," Bioletti said.
"He deals with a lot of customers. He's not a hands-off operator ... I think a charge of that nature definitely has the ability to impact somebody that's fronting the business."
The court also heard Zajmi was in the process of purchasing another business in Mission Bay, had six-figure borrowings and a mortgage to service.
He and his wife employed at least 13 people and were taking on at least three more, Bioletti said.
But Judge Josephine Bouchier was not convinced the threshold for suppression had been met.
Citing previous court decisions, she said: "Suppression should not be used as a tool to protect commercial interests."
The judge also made reference to the fact there were six civilian witnesses who were going to be called by the police to give evidence, who may disseminate the defendant's name.
"On what's in front of me I can't form the view the requisite standard of extreme hardship is met," she said.
The maximum penalty for assault with intent to injure is three years imprisonment.
