Heavy rain caused qualifying to be washed out on Thursday meaning a backlog of matches ahead of Monday's start to the first Grand Slam of the year.
Former US Open juniors champion Watson (pictured) needed just 98 minutes to complete a 6-3 6-2 win over talented French youngster Kristina Mladenovic, while second seed Keothavong was similarly emphatic in posting a 6-1 6-2 win over Darya Kustova of Belarus.
The 18-year-old Watson will next face Ukraine's Lesya Tsurenko while Keothavong will take on Ioana Raluca Olaru of Romania.
But Katie O'Brien threw away a one-set lead as she crashed out 6-3 5-7 3-6 at the hands of 16th-seeded Luxembourger Mandy Minella.
Former world number 22 Sabine Lisicki snuck through a tight clash with Dutchwoman Michaella Krajicek 7-6(2) 6-3 while top-seeded American prospect Coco Vandeweghe battled past Japan's Aiko Nakamura 7-5 6-3.
In the men's draw, Frenchman Nicolas Mahut, seeded 18th, was forced to recover from a set down to book his place in the final round with a 6-7(6) 7-5 6-3 win over Spaniard Guillermo Olaso, while fifth-seeded Marco Chiudinelli of Switzerland lost 6-4 3-6 1-6 at the hands American Alex Bogomolov Jr.