After beating American Allie Kiick at qualifying number two Katie Boulter is one win.
Boulter, 2-2, won 7-5 5-7 6-2 victory within 2 weeks and 19 minutes.
Will face Switzerland’s Viktorija Golubic at the qualifying round on Tuesday.
World number 113 Watson, whose only two wins this year came in January in Hobart qualifying, saved three match points and led 4-2 at the past.
However she also lost four matches in a row because 2012 French Open finalist Errani secured victory.