The duo exchanged several additional breaks of serve before Williams struck her fifth double fault of the day to trail 3-4.
Williams, who had yet to hit an ace for the day, then raised her level of play to break serve tolevel the set at four games all.
Then in a hit or miss, perfect example of the Americanís extreme power game, Williams struck two aces as well as two double faults in her following serve game to hold serve to lead 5-4.
Serving to stay in the match, 199th ranked Beygelzimer trailed 15-30 before facing double match point at 15-40.
Williams, who had struggled to break serve for the day, broke serve on the following point to close out the match.
A good finish this week for Williams, playing in her final scheduled WTA Tour tournament of 2015, could result
in the American reaching the WTA Finals tournament which is to be played the week of October 26th in Singapore.