Higgins was originally scheduled to play against Ronnie O'Sullivan in the first round.
But Robertson joined the tournament in the 11th hour after 'The Rocket' withdrew on medical grounds.
Robertson reaches the quarter-finals of The Masters in extraodinary fashion, coming all the way back from 5-1 down to win 6-5.
In the evening session, England's Shaun Murphy beat compatriot Gary Wilson 6-3.
It was a fairly straightforward job for the world number seven, who led 4-0 before Wilson clawed back three frames.
At 4-3, Murphy claimed back-to-back frames to win and progress to the next round.