Andreeva through in Madrid Open on 16th birthday while Gauff crashes out
World number six Coco Gauff, however, suffered a surprise third-round defeat at the hands of Spaniard Paula Badosa, losing 6-3, 6-0.
Russian teenager Andreeva, ranked at a lowly 194, and who ousted 2021 US Open finalist Leylah Fernandez and world number 14 Beatriz Haddad Maia earlier in the tournament, will face world number two Aryna Sabalenka in the last 16.
Andreeva broke Polish player Linette, world number 19, for 2-0 in the first set, and although the 17th seed broke back, the Russian did it again for a 4-2 lead and served it out.
The youngster broke again in the first game of the second set and converted her second match point to break Linette again and clinch victory.
Andreeva, a wildcard, became only the third 15-year-old to win a WTA 1000 main draw match when she beat Fernandez on Wednesday.
She finished as runner-up at the junior Australian Open in January, while her next opponent Sabalenka won the women's title for her breakthrough Grand Slam.
The Belarusian beat resilient wildcard Camila Osorio 6-4, 7-5 earlier Saturday.
"It was another tough battle, I expected it, it was a really great win, I'm really happy with the tennis today and this win," said Sabalenka.
Gauff, one of the two players who achieved the same feat as Andreeva at 15, was erratic in her defeat by Spanish number one Badosa.
The 19-year-old American, who was runner-up at the French Open last year, broke Badosa in the first game, but the Spaniard battled back immediately to get back on serve, before breaking Gauff again for a 3-2 lead.
Gauff's forehand betrayed her as Badosa broke to love to win the first set 6-3.
The second was a rout with Badosa winning 75 per cent of points.
"I think I served very well, I was playing very aggressively from the baseline, tactically I think I played perfectly and I'm very happy to beat a player like Coco," said Badosa.
"I'm working very hard and it's moments like these which make it worth it."
Badosa will face world number nine Maria Sakkari or compatriot Rebeka Masarova in the fourth round.
Tsitsipas edges Thiem
Stefanos Tsitsipas, the 2019 Madrid runner-up, cracked 35 winners to defeat former world number three Dominic Thiem 3-6, 6-1, 7-6 (7/5).
Thiem, now ranked 93 in the world after a lengthy battle with a wrist injury, battled gamely but succumbed on the Greek's fourth match point.
In a remarkable serving performance, fifth-ranked Tsitsipas hit 39 straight first serves from late in the opening set to early in the third.
"That's fantastic, 39 is a lot," Tsitsipas said. "That's a positive mark for me. I think it's just pure relaxation and focus."
Earlier, world number three Daniil Medvedev saw off Andrea Vavassori 6-4, 6-3 to reach the third round of the men's competition.
Vavassori battled his way past Andy Murray in the first round but could not oust another former world number one in the Spanish capital.
The Italian, ranked 164th, held his own against the second seed but eventually succumbed, even though his serve and volley game troubled Medvedev.
"I think I played a very good match, solid, and saved some breakpoints," said Medvedev after his first match at the tournament.
"I managed to use my opportunities, I felt good and I'm looking forward to my next matches."
Dusan Lajovic earned a surprise win over world number nine Felix Auger-Aliassime, 6-2, 3-6, 7-6 (7/5).
The Serbian stunned 22-time Grand Slam winner and world number one Novak Djokovic on the way to winning the Banja Luka title last week, as well as beating world number six Andrey Rublev in the final.
"Today in the match I really felt like I was going in to show my game," said Lajovic, ranked 40th.
"I knew I would have chances if I played my best tennis. It's really a good feeling to have this much confidence, but I need to be also very careful because it is so easy to lose confidence, especially in tennis."
