NEW YORK -- They were just two particular points from Serena Williams' latest three-set comeback at the US Open, yet they were pivotal and consisted of the sorts of lengthy exchanges filled with athleticism and brilliance that she has been known for throughout her career.

Williams needed both of these points to reverse a deficit of being down a set and a break after 45 minutes of her quarterfinal against Tsvetana Pironkova on a cloudy Wednesday in empty Arthur Ashe Stadium.

She needed both of these points during a five-game, match-altering run -- along with her 20 aces -- to end up on the right side of a 4-6, 6-3, 6-2 score after more than two hours to get to the semifinals at Flushing Meadows for an 11th consecutive appearance.

"I never give up," Williams said. "I've just got to keep going."

Two more victories would allow her to claim a record-tying 24th Grand Slam singles title.

But she'll want to avoid another slow start.

"In the beginning, I was a little fatigued, for whatever reason," Williams said. "Obviously I can't do that if I want to keep winning, so I need to figure that out."

How big an upset would this have been if Pironkova had held on? She is not seeded at Flushing Meadows and doesn't even appear in the WTA rankings -- this was her first tournament of any sort in more than three years because she left the tour to become a mother.

"It's unbelievable," Williams said about Pironkova's impressive return to competition. "Wow. I couldn't even do that."

Indeed, when the players stepped onto the court, the stadium announcer -- announcing for whom, exactly, was something of a mystery -- referred to Pironkova, a 32-year-old from Bulgaria, as "Alexander's mom" and then to Williams as "Olympia's mom" during the prematch introductions.

"It just shows me how tough moms are. Whenever you can birth a baby, honestly, you can do anything," said Williams, whose daughter turned 3 on Sept. 1 and is a little older than Pironkova's son.

"You play a match and you go home and you're still changing diapers. It's like a double life. It's really surreal."

Williams, who turns 39 in less than three weeks, has won six championships at the US Open and was the runner-up four times, including in 2018 and 2019.

Williams last lost before the semifinals in New York all the way back in 2007, when Justine Henin eliminated her in the quarterfinals.

On Thursday, Williams will face No. 16 Elise Mertens or unseeded Victoria Azarenka. Their quarterfinal was scheduled for Wednesday night.

This week, in the fourth round, Williams also needed a comeback and the maximum number of sets before defeating 15th-seeded Maria Sakkari 6-3, 6-7 (6), 6-3.

In Wednesday's turnaround, the first key moment involved 24 strokes, the next-to-last a running crosscourt forehand by Williams from wide of the doubles alley and the last an on-the-run squash forehand by Pironkova that landed in the net, giving Williams a break and a 5-3 edge in the second set.

Williams raised her left fist as her husband yelled from his front-row corner seat; Pironkova put a hand on her knee, smiled ruefully and squatted behind the baseline.

The other pivotal moment came in a four-deuce opening game of the final set and began with right-handed Williams taking a page out of Maria Sharapova's playbook by hitting a left-handed return of serve. Another 15 strokes followed, with Williams hitting a forehand passing shot that Pironkova volleyed into the tape of the net.

Pironkova dropped onto her back, chest heaving; she left so much sweat on the court that a ball person was beckoned to wipe it away with a towel. That stretch afforded Williams a third break point, which was converted for a 1-0 lead when Pironkova sent a forehand long.

"She definitely played like the champion she is,'' Pironkova said.

Williams then only added to the lead, her strokes finding targets better as the match went on. Her serve was especially good.

In contrast, Pironkova began to have more trouble on the longer points that she dominated early; she won the first half-dozen that lasted 10 strokes or more before Williams found more success.

A couple of times, Pironkova drew Williams forward with a drop shot, then took the point with a perfectly arced lob.

"I tried everything,'' Pironkova said.

But after compiling only five unforced errors in the first set, Pironkova had 21 over the last two. Williams, in contrast, made 11 in the first set, just 13 the rest of the way.

"Definitely, the first set, I think I was in control of the match. I was doing all kinds of shots, and everything went ... my way,'' Pironkova said. "But, you know, I was expecting (that) it's not going to last forever.''