The two of them decided they did not want to be parted, he got a divorce and they embarked on a tender 12-year marriage cut short only by Bogart's untimely death from throat cancer.
That said, neither of them went on national television and jumped up and down on a studio couch to proclaim their feelings for each other. It was a different age, in which the private lives of stars were certainly talked about in the press but the stars themselves did not discuss their intimate feelings in public.
All that might explain why Bacall, now 80 and still on the big screen, has little patience for the antics of Tom Cruise and his new, much younger, very public fiancée Katie Holmes.
"His whole behaviour is so shocking," Bacall told Time magazine over the weekend. "It's inappropriate and vulgar and absolutely unacceptable to use your private life to sell anything commercially. I think it's a kind of sickness."
Bacall is hardly the first person to pour scorn on the way Cruise and Holmes have squeezed their relationship for every last drop of promotional juice, at a time when both of them have big summer blockbusters to push. Cruise, arguably the biggest male star in Hollywood, has become something close to a figure of ridicule for his unconvincing - and repeated -- protestations of undying love, his insistence on bringing his Scientology religion into every aspect of his professional life and his habit of alerting the media every time his wedding plans with Holmes take an extra step forward.
But it is unusual for a veteran Hollywood figure such as Bacall to take a potshot at a younger colleague. In her Time interview, she stated point-blank: "When you talk about a great actor, you're not talking about Tom Cruise."
Her views might or might not be coloured by the fact that she has co-starred twice with Cruise's former wife Nicole Kidman, in Lars von Trier's Dogville and now in Jonathan Glazer's Birth, in which she plays Kidman's mother. She told an interviewer at the Venice film festival last year that she and Kidman developed a strong friendship.
But that did not stop her from bridling at the description of Kidman as a screen legend. "She's not a legend. She's a beginner," Bacall said. "She can't be a legend at whatever age she is."
When Bacall was making her start, in classics including To Have and Have Not, The Big Sleep and Key Largo, the off-screen chemistry was left largely to audiences' imagination. When Bogart as Philip Marlowe tells her: "You've got a touch of class, but I don't know how far you can go," she responds: "A lot depends on who's in the saddle." But these days, the off-screen behaviour of stars has almost entirely eclipsed what goes on in their movies.
Neal Gabler, author of the book Life The Movie: How Entertainment Conquered Reality, wrote over the weekend: "Watching a movie used to tickle viewers to want to know more about its stars. Today, knowing about the stars is an end in itself." Gabler added: "Movies have become what director Alfred Hitchcock called a 'MacGuffin', a red herring that triggers a plot but has no other inherent value. Like MacGuffins, movies have little inherent purpose except to be talked about, written about, learned about, shared as information."