Former two-time champion Williams defeated Linette 6-3, 6-7 (6/8), 6-2 to reach the quarterfinals of the hardcourt tournament at Stanford University.
The 36-year-old Williams is hoping she doesn't suffer a post-Wimbledon letdown and she now has an easy path to the semifinals since her next opponent won't be ranked in the top 200 in the world.
Williams will face either Catherine Bellis, who is ranked 203rd, or Sachia Vickery who is one spot back of Bellis in the world rankings.
"It would be very special to get No 50 here but I am just going to try to keep advancing," said Williams who has 49 career titles after winning in Kaohsiung, Taiwan earlier this year.
Williams blasted 11 aces and had four double faults against Linette in the two hour, 14 minute match on the main stadium.
Earlier this month Williams reached the semifinals of Wimbledon, her first semi at a major in five years.
Williams played her first professional tennis match at this event in 1994 and she has won here twice but not since beating Kim Clijsters 14 years ago.
The former world No 1 has seen her game fall on hard times over the past several years.
And she got off to a rocky start to this season before recovering, losing in the first round of the Australian Open to the 47th ranked Johanna Konta.
To add insult to injury, organisers fined her a record $5 000 for refusing to attend the post-match news conference. She also suffered losses in opening second round matches at both Miami and Indian Wells.
Linette, who is ranked second in Poland behind Agnieszka Radwanska, had to be pleased to take the world No 7 to three sets in her Stanford debut.
In 2015 she made her Grand Slam debuts at the French Open, US Open and Wimbledon.
