The movie, which stars Chris Pine, Zoe Saldana, Benedict Cumberbatch and Zachary Quinto, is a follow-up to the 2009 film based on the long-running Star Trek television series.
Abrams extended an invitation to a terminally ill man identified only as 'Dan', whose friends had appealed for a chance for him to see the movie via an internet campaign.
Dan, who is described as a huge fan of the Star Trek series and a movie buff, has been suffering from two forms of cancer over the past three years. He was recently informed that he has only weeks to live, according to a post made by a friend on Reddit.
Abrams became aware of the internet campaign and reached out to Dan personally, reportedly inviting him and his wife to see an unfinished version of Star Trek into Darkness at a special private screening.
Photo gallery - 'Star Trek' - The movie series in pictures:wrote: "Hi everyone, it's Daniel's wife, we are of course sworn to secrecy, but we are officially allowed to say we saw it and we enjoyed it immensely as a film and as a gesture... Also, it was truly amazing that a filmmaker so secretive as JJ Abrams was kind enough to show this to us.
"It is also so wonderful for his friends and his family to finally hear some good news from us, they know how important [the] film is to Daniel and could not think of a better way to please him. So this gesture really has brought a lot of smiles all around. It was a wonderful thing to see with Daniel and a wonderful thing to see Daniel enjoy - Making someone as ill as he is smile for any length of time really makes a difference. (At this point making me smile is good too).
She went on to thank Abrams, saying: "We were so touched when [Abrams] was concerned that we wouldn't 100% enjoy the screening as it's still being worked on... Be assured we enjoyed it 110%!"
Star Trek into Darkness is set for release on May 17, 2013. Watch a trailer for the movie below: