Written by Mark Gatiss, An Adventure in Space and Time features David Bradley as Hartnell, the first actor to play the Doctor in 1963.
Jessica Carney, who was six when her grandfather took the role, said he would have been “thrilled” by the show.
She was clearly overcome with emotion after a screening in London on Tuesday.
Some fans had queued up for hours for return tickets to the preview, held at BFI Southbank, which was attended by a number of former Doctor Who stars.
Afterwards, Carney thanked Bradley “for playing my grandfather so wonderfully”.
An Adventure in Space and Time is set in the first half of the 1960s and focuses on the people who created the sci-fi show, which celebrates its 50th anniversary next week.