The clip was promised to fans of the show if details of the series finale 'The Name of the Doctor' were kept a secret. The series Blu-ray was accidentally leaked to some fans in the US last week.
The official Doctor Who Twitter account tweeted earlier in the week: "Steven Moffat has promised if fans help keep the finale's secrets, we'll release a special video featuring Matt & David right after the ep!"
In the interview, Smith and Tennant discussed the first time they met on the set of 'The End of Time' in 2009, as well as the "unusual set of pressures" that taking the role brings.
Smith noted that the Tenth and Eleventh Doctors seem to "get on quite well" in the anniversary episode, while Tennant added that they do "bicker a bit".
The special episode will air on November 23 and will also star John Hurt, Jenna-Louise Coleman and Billie Piper.
Watch Digital Spy's Alex Fletcher and Morgan Jeffery discuss 'The Name of the Doctor' in 'Geek TV' below: