After splitting the first two sets, which lasted a combined total of an hour and 50 minutes, it was the No.7-seeded Williams who seemed headed for a more routine victory early on in the third set, pulling away from 1-all to build a 5-3 lead. Zahlavova Strycova wasn't done - she won three games in a row to go up 6-5, and in that game she even brought up a match point with Williams serving at 30-40.
But Williams summoned up one last burst of energy, winning three points in a row to dig out of that game and then sneaking out a close tie-break to finish Zahlavova Strycova off, 7-5, 3-6, 7-6(5).
Williams is now through to the Qatar Total Open quarterfinals for the first time, and she's actually the first player into the quarterfinals - all the other matches on Tuesday were first round matches.
The two seeds playing their first round matches on Tuesday both advanced, with No.6 seed Andrea Petkovic rallying to beat Belgian qualifier Kirsten Flipkens, 6-7(5), 6-3, 6-2, and No.9 seed Carla Suárez Navarro winning when Garbiñe Muguruza retired down 6-5 first set due to a viral illness.
Jelena Jankovic, Romanian qualifier Alexandra Dulgheru and a trio of WTA Rising Stars - Karolina Pliskova, Elina Svitolina and Zarina Diyas - rounded out the winners. Fresh off the biggest final of her career at the Premier 5-level tournament in Dubai last week, Pliskova battled hard for a 7-6(4), 6-4 victory over Stefanie Voegele, even fighting off a set point down 6-5 in the opening set.
Pliskova was asked afterwards whether winning so many matches is taking its toll.
"Well, I had Sunday off - we came from Dubai," Pliskova said. "And on Monday I went to hit just once, because I didn't feel really well, I was a little bit tired from the whole week in Dubai. So I just hit like 40 minutes with my coach and today I just warmed up before the match, so it wasn't that much.
"I'm tired, but I'm still in the tournament, so I don't want to think about it. I'm thinking about my next match and will do my best there, and then I'll take some time off after the tournament."