The officer said the passenger was overjoyed when the car stopped.
"He thanked me with very colourful language," he added. "Let's just say he was very grateful."
PC Smith, who was back in work yesterday afternoon, added: "It's one of the first times I've used these new vehicles and I was amazed at how well my car responded."
Mr Smith's commanding officer, Inspector Dave Corcoran, said the modest officer had performed an excellent job, and he would be recommending him for an award.
"I don't think Adrian appreciates the fact he has undoubtedly saved many lives," he said.
"He has prevented a major incident. I'll be recommending him for a commendation. I'm very proud of what he did."
The patrol car was hardly damaged but the Fiat was badly smashed in the incident.
PC Smith treated the driver, a 51-year-old from Ireland, at the scene until paramedics arrived to rush him to the Royal Liverpool Hospital. It emerged later that he had been suffering from hypertension, and he was released yesterday evening after treatment.
Leading traffic officer Sgt Paul Lynch said: "This was an extremely courageous act by Constable Smith, whose quick and selfless actions undoubtedly saved this man's life and potentially the lives of many others."
Traffic was diverted to the Kingsway Tunnel, before the tunnel reopened at 10am.