A stand-up star for more than 50 years, Newhart became something of an overnight sensation for a new generation of followers with a couple of cameo performances in The Big Bang Theory, a comedy about young science geeks.
He recently picked up his first Emmy (after six previous nominations; the first was in 1962) for his role as Professor Proton, a faded, ageing TV scientist, in the comedy that has become the most-watched non-sports show on a digital channel.
Speaking from his home in Los Angeles, he tells me the long-awaited Emmy, for Oustanding Guest Actor, was very dear to him. "It's a great award because it comes from your peers and that has more meaning," he says. "What was especially moving about the Emmy award was that there was a standing ovation. That was very special; that what your peers thought you were doing merited that."
But how did a seemingly self-effacing, mild-mannered comedian, famed for his slightly stammering delivery in 1960s monologues like The Driving Instructor and Introducing Tobacco to Civilisation and sitcom the Bob Newhart Show in the 1970s, find himself a foil to Leonard and Sheldon on TBBT?
"Producer Chuck Lorre had been approaching me over the years for various appearances and for one reason and another I couldn't do them. So in 2013 he came to me and said: 'I'm here for my annual turn-down'.
"I said: 'I'll tell you what: I like Big Bang Theory, I like the writing, I like the cast. I'd like to do it, but I need to do it in front of a live audience, because that's the only way I can perform. I tried it once without a live audience and I find it very sterile and difficult. Having come up though stand-up, my whole life has been relating to an audience."
Newhart's wish was granted and the rest, as they say, is history. He has just filmed a third appearance (it airs in the US on May 1) and there is big news: "The next one will indicate a kind of a change which I'm not free to divulge. That will probably mean more appearances on The Big Bang Theory."
That's a pleasing prospect. He says of his young castmates: "I couldn't have felt more at home. It's difficult when you've been on your own stage for many years to walk on to somebody else's. They couldn't have been nicer, making me feel at home. They are extremely talented and certainly entitled to all the success they have attained."
Did taking on a new challenge at the age of 84 raise any alarm bells among his family (he has been married to Ginny for 51 years; they have four children and nine grandchildren)? "I once did a three-parter ER and in it I committed suicide - it got me an Emmy nomination. The kids enjoyed it, but they said:'You can do these shows, but don't do anything like that, OK?' I said: 'OK, I promise'.
"But I'm not looking to go back on television full-time. It's extremely demanding and at my age it would be physically impossible. It's emotionally draining when it doesn't work week after week, so I'm not looking for a regular series."
He does, though, keep himself busy with his stand-up career, although he's never been a quickfire gag-after-gag performer. He is often, uniquely, his own straight man; his greatest routines involve him holding down one end of a conversation, subtly allowing the imaginary second party to provide the comedy. "I've just celebrated my 54th year in stand-up. I can't imagine not doing it. It keeps my mind active and gives me something to look forward to. As long as I am physically able to, which I seem to be at the moment, I can't ever imagine not doing stand-up. I love working in front of a live audience. I just recently did three shows in Florida and they went wonderfully."
He keeps his eye on today's performers, too: "Among the ones working just now, aside from the obvious like Jerry Seinfeld and Ray Romano, I like an American comedian called Jim Gaffigan and also Jake Johannsen. They are extremely creative and they work clean, which I like.
"But my all-time favourite stand-up was Richard Pryor. His work was incredible; take away the language and the nugget of the comedy is just amazing."
Now well into his sixth decade as a comedy star, with a new raft of admirers on both sides of the Atlantic, it has to be said that Bob Newhart is pretty amazing, too.
The Big Bang Theory, E4, Thursdays, 8pm.