The Scot is bidding to overhaul Novak Djokovic as world number one and needs to at least reach the final in Paris if he is to stand a chance of topping the rankings on Monday.
Murray looked somewhat lethargic against Pouille but went on to win eight successive games after the world number 17 had broken the Briton's serve midway through the first set.
The Wimbledon champion then broke three further times in the second set to win the match and cruise into the quarter-finals.
It was a 17th win in a row for the 29-year-old, who is aiming to win a fourth straight title after winning trophies in Beijing, Shanghai and Vienna last week.
Murray would also become the first British player to be ranked number one in singles should he topple Djokovic.