The longer he was onstage, and the more he sang, the more animated he became - and it seemed that the enthusiastic response from the audience was fuelling his staggeringly lively performance.
But it seems that there is more to it than that. Bennett, who is now 88 and set to return to the Concert Hall in early September, is not so much driven by the need for applause as he is by the desire to entertain, and by his own enormous pleasure in singing. "I love doing it," he says, "and I like to try to make people feel good. It's always very enjoyable to me leaving the theatre knowing that I made people feel good."
And it's not just to his audience that Bennett feels a sense of responsibility; as the oldest popular singer on the block - and the one whose career stretches back over an incredible six decades - he regards himself as a custodian of the Great American Songbook. After all, there can only be a handful of singers still around who have direct links with the original contributors to that body of work, and such original exponents of it as Bennett's hero, Fred Astaire. Does Bennett feel a sense of responsibility to these songs?
"Yes, I do," he admits, "because the United States in the 1930s had a renaissance period very similar to what happened in France with Impressionism, with Monet, and musically with Ravel and Debussy. It was the beginning of talkies in films so they grabbed Fred Astaire off the stage and put him in films, and they hired George Gershwin and Irving Berlin and Cole Porter to write the songs. These songs are gorgeous; they never become dated because they're so well-written. I travel the world and wherever I go people start to sing them back at me - they're known internationally."
Bennett first heard many of them - including the one he cites as his personal favourite, Jerome Kern's All The Things You Are ("I just adore that song") - as a youngster growing up in the Queens borough of New York. Before he discovered this homegrown American music, however, it was his father's Caruso records which introduced him to the art of singing, and in particular to the "bel canto" style, which explains his graceful way with a song and his elegant phrasing.
"My father adored opera and had a reputation himself as a singer. I was told that he would sing on the top of a mountain in Calabria and the whole valley would hear him. This inspired my older brother and myself - and we both became singers. My brother was very successful - at age 14 he was hired by the Metropolitan Opera House. So he was called Little Caruso. Of course I became a bit envious so I just became interested in jazz and started improvising."
Asked who his first inspiration as a singer was, Bennett instantly names Louis Armstrong who, like him, enjoyed success as both a jazz and a popular artist, and who effectively invented jazz singing. That the pair became great friends is perhaps no surprise given that they seem to share the same outlook about entertaining an audience and living life to the full. "His whole life he just wanted to make people feel good and have fun," Bennett says. "He loved what he was doing so much that it never became old-fashioned. Just listen to him playing on a Hot 5 record. If you listen to the musicians playing behind him, it does sound a little dated, but when Louis comes in on trumpet or singing it sounds like right now."
Speaking to Tony Bennett, it's impossible not to be struck by his delight in discussing jazz - his first musical love - and its characters.
On Duke Ellington: "He was a complete genius, unbelievable. He just performed every night. I knew him at least the last 30 years of his life and there wasn't a day that he didn't compose some music. Even when they were on tour doing one-nighters and he was travelling 150 miles a day, he would have the orchestra try something that he might want to put into his composition."
On John Bunch, the much-loved pianist and former music director to Bennett, who was a regular visitor to Scotland's jazz festivals until his death in 2010: "Gentleman John Bunch. I loved him so much. He was the most wonderful person. In fact, I'll tell you a cute story ... He asked me one time when we were in London: 'Did you ever play tennis?' I said 'No.' He said, 'Would you like me to give you a lesson?' I said 'Okay.' So he took me out to a stadium to play at the net and he showed me how to hit the ball over the net and all that sort of thing. Later on I found out that we had been playing at Wimbledon! I've been working down ever since then…"
Billie Holiday ("a sweet, beautiful, sophisticated lady") was a particular favourite - and Bennett, who first topped the charts in 1950, was lucky enough to meet her. "Duke Ellington had a show at a nightclub in New York and I went to see it. Billie Holiday was there too. It was the days when there was an awful lot of prejudice. She said: 'C'mon Tony, let's go uptown and have a jam session.' The people I was with kind of indicated to me 'don't go up there - it's dangerous,' you know? I regret it to this day."
Bennett may be embarrassed about that episode but it was exceptional, since he was an active supporter of and participant in the civil rights movement in the 1960s, responding to the call to arms from his friends Martin Luther King and Harry Belafonte. His own life hadn't been without struggle, and the other overwhelmingly striking thing about a conversation with him is how lucky he feels to have had the life he's had, making a living doing something he loves, and how - even now, nearly 60 years after his initial success - he still counts his blessings.
"After my father died [when Bennett was 10 years old], my Italian family would come over every Sunday and my brother, sister and I would entertain them. They told me that I sang very well and that they liked my paintings of flowers. They created a passion in my life to just sing and paint, and I've gotten away with it - I've never really worked a day in my life. I just enjoy what I do." Asked if he escaped into his music when times were tough, Bennett explains that it wasn't merely a form of distraction; it was a practical escape route out of poverty. "I went into showbusiness to stop my mother from working - she was making a penny a dress sewing in a sweat shop to put food on the table for her children. I was able to accomplish that with my first couple of hit records - I was able to send my mom out into the suburbs into beautiful nature."
The adult Bennett has had his fair share of professional frustrations and personal problems - and for a while he was, to paraphrase one of his early signature songs, "lost, his losing dice was tossed, his bridges all were crossed, nowhere to go".
A period out of the pop limelight in the 1970s produced some jazz albums - notably duets with pianist Bill Evans and, most sublimely, his two volumes of the Rodgers & Hart Songbook with the Ruby Braff/George Barnes Quartet; intimate, cult recordings which are among the very best vocal works in all jazz and which highlight Bennett's admiration and respect for some of the most eloquent lyrics in the Great American Songbook.
Relaunching his career in the late 1980s, performing on MTV, and duetting with young pop stars - most recently Lady Gaga - has brought him to new audiences. But best of all, his later success has allowed him to be exactly the kind of singer he wants to be - singing jazz with his quartet and creating an intimate atmosphere even in the largest venue.
"I like working that way," he says. "To clarify my whole premise: I don't want to be the biggest. I'd rather be one of the best." Mission accomplished, I reckon.
Tony Bennett performs at the Glasgow Royal Concert Hall on September 9, www.glasgowconcerthalls.com