"So I have just got to keep saying at each tournament that my goal is just to be better than the last". It was one of Serena's 41 unforced errors, including 17 on the forehand side.
Venus saved a break point of her own at 5-4 and went on to take the match.
This is the first time the pair have faced off since Serena beat Venus in the finals of the Australian Open in January 2017.
Venus will now be up against 21st seeded Anastasija Sevastova of Latvia in the fourth round of the tournament. "I think it's such a treat for everyone to see this match so early in her comeback", Venus said.
However, she revealed in February that she "almost died" after giving birth and was bedridden for six weeks.
"I haven't played in over a year; it's definitely not less disappointing", she said of the defeat.
This was the first singles tournament of her comeback, and she arrived with no official ranking or seeding, resulting in this third-round clash with the eighth-seeded Venus.
Wozniacki said the court conditions were hard but she managed to make the adjustments. "I don't want to go backwards". "My room for improvement is incredible", added Serena.
The 23-time Grand Slam champion hit with power; and she got better as her first two matches progressed.
Meanwhile, the 37 year old Venus Williams has leaped to the quarterfinals of the Indian Wells tournament where she will be meeting the 27 seeded Spaniard Carla Suarez Navarro.
Sixteen-year-old American Amanda Anisimova saw her surprising run end with a 6-1, 7-6 (2) loss to fifth-seeded Karolina Pliskova of the Czech Republic.
The Indian Wells match was 17 years in the making. "I thought I was in control of the point and winning the point, and she turned it around".
That triggered ugly scenes, as their father Richard said he was racially abused by fans when Serena then played - and won - the final against Kim Clijsters. "They are both over 35, 36, and still basically playing at the top with Serena being a mom now and Venus overcame a lot of also hard situations and still playing". "But she definitely played a little better than she normally does today".
Asked if playing Monday brought closure to what happened in 2001, seven-time grand slam champion Venus replied: "It never crossed my mind".