Konta, the British number one, beat Colombia's Mariana Duque-Marino 6-3 6-3, while Watson saw off Japan's Misaki Doi 6-4 6-7 (0-7) 6-3.
Australian Open semi-finalist Konta will play either Belgian Yanina Wickmayer or Johanna Larsson of Sweden.
Watson, who has fallen to 84th in the world, faces either Hungarian Timea Babos or Slovenia's Polona Hercog.
Fourth seed Konta, 24, broke once as she took the opening set in 32 minutes.
After dropping serve immediately after breaking in the second, she reeled off four of the next five games to seal her place in the next round.
