Robertson, 27, scored the second goal in GB&NI’s 4-3 victory in a superb match and also laid on the winning counter in the final quarter.
With eight and a half minutes gone in the second quarter, Hannah Martin passed to Shona McCallin, who crossed from the right for Robertson to collect the ball, swivel round and hit a fierce backhand shot in off the post.
She ended up off balance on the ground and was smothered by her team mates in celebration.
That increased the UK lead to 2-0, after a deflected cut-back by Ellie Rayer had found the net within the opening 30 seconds of the quarter.
But India, shock winners over Australia in the last eight, struck back as Gurjit Kaur scored from two penalty corners in close succession and Vandana Katariya made it 3-2 after squeezing the ball home in a crowded D.
GB&NI captain Hollie Pearne-Webb, however, signalled the fightback with a fierce strike from distance. Then, two minutes into the last quarter, Robertson administered the injection from a penalty corner which enabled Grace Balsdon to fire home the decisive goal.
GB&NI had been the defending champions from the 2016 games in Brazil and, despite having a few survivors from that squad, were keen to see how their new-look modern squad would blend together in Japan.
The British women were also bronze medallists at London in 2012.
The gold medal decider in Tokyo 2020 takes place later today between Holland and Argentina.