Robson is just one victory away from her first senior French Open appearance after she beat Laura Thorpe in the second round of qualifying.
The British number four overcame her French opponent in a tight encounter 7-5 7-6(5) to reach the third and final round of qualification matches for the only Grand Slam in which she is yet to compete.
The 18-year-old, who won the Wimbledon Junior Girls' Championship in 2008 and twice reached the final of the Australian Open in that age bracket, must now defeat Karolina Pliskova of the Czech Republic if she to win a place in the main draw in Paris.
Watson, 20, defeated Ons Jabeur of Tunisia 7-6(5) 6-3 and will face Anna Floris of Italy in the final qualifying round on Friday.
Floris beat Leticia Costas-Moreira of Spain 6-2 6-0.
Watson reached the tournament proper at Roland Garros in 2011, becoming the first British woman in 28 years to make it through qualifying there. She then became the first to reach the second round since 1994 by beating Stephanie Foretz Gacon.
On Thursday she was cruising at 5-2 up in the first set only to be pegged back to a tie-break by reigning junior champion Jabeur. She found herself 5-1 down in that breaker before putting together a comeback of her own, featuring six points in a row, to seal the set.
The second set was a more straightforward affair, Watson claiming it 6-3 from a position of three games all.