Smoking wasn't nearly as stigmatized in the 1960s as it is in the 21st century, not even among the astronaut corps. Many of the early astronauts smoked, some of them through their entire careers with the space agency. They were not required to quit, and to my knowledge none of the early astronauts has ever expressed a feeling that they were under pressure to do so.

Missions from the founding of NASA in 1958 through to the last Apollo mission in 1972 ran for no longer than two weeks, and the simplest answer to your question is that on longer missions they were just expected to control themselves and go without for the duration. The cabin atmosphere on all of the Mercury, Gemini, and Apollo missions was comprised of 100 percent pure oxygen after launch, and even aside from the irritation that the smoke would cause, lighting a cigarette in this environment was a profoundly bad idea, as exemplified by the intensity of the accidental Apollo 1 launch pad fire in 1967 that killed three astronauts.

The lack of pressure to quit smoking is well expressed by Apollo 15 astronaut Al Worden in his 2011 memoir Falling to Earth:

"Many people assumed that astronauts needed to be in top physical condition, and that NASA had some kind of exercise plan we all had to follow. Not true. Our bosses believed we were grown-ups who knew we had to stay in shape, and they allowed us to follow our individual sports and exercise pursuits. Ironically, we had some pretty unhealthy habits. For example, just about everybody smoked back then. It was not frowned on like it is today.

I'm still smoking today. In fact I am probably the only person in the whole goddamn program who hasn’t given it up. I have managed to kick the habit for a couple of years at a time, but never completely. I smoked all the way through NASA, and so did almost everybody I ever shared an office with. If I could have worked out how to do it safely, I’d have smoked all the way to the moon and back."

Gordon Cooper, one of the original seven Mercury astronauts, also mentioned the prevalence of smoking in the early days in his memoir Leap of Faith:

"There's a reason I used less oxygen [on my mission] than any of my predecessors: I was the first lifetime nonsmoker in space. Some, like John Glenn, quit during the program; others, like Wally Schirra and Al Shepard, smoked right up to the launch pad. In that controlled environment it was documented that my lungs were much more efficient. My oxygen consumption was only one-third of what they had calculated it would be based on previous flights, a powerful argument for not smoking."

So a lot of smokers flew into space in the early American space program, but they had to abstain during the missions themselves. This would have been through sheer willpower, these being the days before nicotine patches or nicotine gum. That it sometimes wasn't the easiest thing to do is described by Gene Cernan, the commander of the last Apollo mission to the Moon in 1972. In his memoir The Last Man on the Moon he describes his crewmate Ron Evans just before the launch of Apollo 17:

"Ron smoked the last cigarette he would have for almost two weeks. Its memory must have haunted him, because one of the first things he would do aboard the recovery ship was bum a cigarette from a sailor."

As a postscript, all of this isn't to say that a cigarette has never been in space. Astronauts have taken personal items into space since the very beginning, either officially approved - such as in the crew Personal Preference Kits (PPKs) flown on Apollo spacecraft - or unofficially secreted away in a suit pocket. For example, Gus Grissom famously had rolls of Mercury dimes hidden inside his suit pockets during his Mercury-Redstone 4 launch in 1961.

Of more relevance to this question, Wally Schirra entered his Mercury-Atlas 8 spacecraft in 1962 to find that, as a prank or "gotcha" as Wally called it - the ground crew had stored a pack of cigarettes and a small bottle of Scotch whisky aboard. In another example, Buzz Aldrin, who orbited the Earth on Gemini 12 before he flew to the Moon on Apollo 11, discussed in his own memoir Magnificent Desolation how he took a pipe with him on Gemini 12. He apparently rarely smoked it on his own time, but liked to chew on the end of it.

So, at the end of things, while they may have been up there, it's safe to say that no one aboard an American spacecraft has ever lit up a cigarette in space.

Sources

Buzz Aldrin (with Ken Abraham), Magnificent Desolation: The Long Journey Home from the Moon (2009).

Eugene Cernan (with Don Davis), The Last Man on the Moon (1999).

Gordon Cooper (with Bruce Henderson), Leap of Faith: An Astronaut's Journey into the Unknown (2000).

Al Worden (with Francis French), Falling to Earth: An Apollo 15 Astronaut's Journey to the Moon (2011).