The second-seeded American trailed the fifth-seeded Russian Andreeva by a set and a break before rallying back to complete a two-hour 38-minute win.
She thereby booked a place in the semi-finals of a WTA 1000 event for the fourth time in her career.
"I was almost in tears there at the end," Anisimova said.
"It made me emotional seeing her like that. I feel we both won today," she added.
Anisimova next faces fellow American Jessica Pegula, who earlier beat Dane Clara Tauson 6-3, 2-6, 6-4.
Andreeva took the first set with two breaks and leapt to a 2-0 advantage in the second set before Anisimova retaliated and clinched the next five games.
Anisimova was broken serving for the second set, and it was Andreeva's turn to fight back as she took three games in a row to level.
She broke again to take the quarter-final clash into a deciding set.
Anisimova served for the match at 5-3, but Andreeva sliced and lobbed to stay alive and soon drew level.
Andreeva then served for the victory, but she could not close at 6-5, and the contest went to a deciding tiebreak.
Anisimova upped her level to take the breaker and hugged a sobbing Andreeva at the net, telling her she was "amazing."
Gauff thrashes Eala
In the night session, Coco Gauff made it three Americans in the semi-finals with a merciless 6-0, 6-2 dismissal of Filipina rising star Alexandra Eala.
In front of a buoyant Filipino-majority crowd, Gauff took the first nine games of her match before Eala finally got on the board.
The third-seeded Gauff, who struggled on serve in the previous round, committing a whopping 16 double faults, struck her eighth of the match to give Eala a lifeline and surrender a break.
That was the last blip from Gauff, who wrapped up the win on the 67-minute mark to reach her second Dubai semi-final.
"I think tennis needs more of this," Gauff said, speaking of the raucous atmosphere created by Eala's fans, who have sold out the stadium for her matches all week.
"It's great just to see a nation so proud of someone, especially a nation that's pretty under-represented in this sport. I think it's pretty cool to see.
"I also liked, even at 6-0, 4-1 or 4-0, she was smiling. I think that's something that you can't buy. I see why people want to root for her. I root for her when I'm not playing her, of course."
Gauff will take on two-time Dubai champion Elina Svitolina in the final four. In the last match of the day, the Ukrainian came back to beat Croatian lucky loser Antonia Ruzic 3-6, 6-2, 6-3.
Pegula reaches seventh-straight semi
Pegula advanced to a seventh consecutive Tour-level semi-final - dating back to the 2025 US Open - with a hard-fought three-set win against last year's runner-up Tauson.
"I think I've been serving a lot better. Physically, been feeling good," Pegula said.
"I've been working on my game a lot. I think I've become a better player over the last six months."
In their first meeting, Pegula and Tauson split the first two sets and were neck and neck in the decider until the American claimed a crucial break in the seventh game.
