Price's promoter Frank Maloney, who also guided Lewis's career, said that he is excited about the future after such a devastating performance.
"It was 20 years ago that Lennox knocked out Razor Ruddock, and that punch was better than the one Lennox threw," he said. "David is the future of British boxing."
Price admitted that he was surprised with his punching power and the speed of the victory.
"I didn't expect to do that to Audley Harrison, I thought I could break him down over time," said Price.
"I've developed a killer instinct over the last year or so. When I smell blood, I'll get him out of there.
"Each win brings more confidence and I believe I will be heavyweight champion of the world. There is still massive room for improvement though."
Price's next fight will be against former British champion Matt Skelton in December.