Danilovic continued to be the aggressor in the second set but a couple of miscues allowed her Belarusian opponent to force a decider.
Sasnovich fought back from two breaks of serve down in the final set to level at 5-5 but could not seize the initiative, with Danilovic dominating the rallies to set up a third chance to serve out the match.
A long forehand from Sasnovich gave Danilovic -- who was granted a place despite initially failing to qualify -- victory and a place in the final on Moscow clay against fellow teenager, Russian wildcard Anastasia Potapova, who came back from a set down to take the other semi-final.
Potapova, the 2016 Wimbledon girls' singles champion, struck 26 winners in an hard-hitting display before her Slovenian opponent Tamara Zidansek, who was struggling with illness, retired, trailing 3-6 6-4 5-2.
(Reporting by Hardik Vyas in Bengaluru,; Editing by Neville Dalton)
