The capacity audience of 5,000 was evacuated after smoke was seen as Bennett was singing one of his biggest hits, I Left My Heart in San Francisco.
"I knew I was hot, but not that hot," the star joked from the stage after informing the crowd that there was a fire.
The small blaze was started by a burnt-out motor, the venue said, and Bennett was due to return to play the second of his two shows there on Friday.
"We understand how disappointed the audience were, but the safety of the public is our paramount concern," a spokesman for the prestigious venue said.
"As it turned out, the public were never at risk, but we cannot take chances."
The 75-year-old crooner was "very disappointed" after refurning to his hotel when the show was abandoned.
"He didn't want his audience to be disappointed, but his main concern was obviously for their safety," his spokeswoman said.
One fan, Keith Webb, a retired businessman from Gerrards Cross, Buckinghamshire, who was in a box in the venue, said: "It was all quite orderly, there were no signs of panic or people rushing to leave quickly.
"Downstairs staff said the show had been cancelled, but no one mentioned anything about refunds."
George Walpole, 37, travelled from Dublin to London to see the show, and said: "We got held up in traffic and missed the start, and just as we got here, hundreds of people were pouring out of the building.
"We paid £37 for the ticket and we want a refund."
Bennett, considered to have one of the best voices of any living male singers, is due to finish his short trip to the UK with a show at Liverpool King's Dock on Sunday, 7 July.