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 now gets a rematch with fellow American Jessica Pegula.
Two breaks gave Andreeva a one-set lead, and the defending champion leapt to a 2-0 advantage in the second set before Anisimova retaliated and clinched the next five games.
Anisimova got 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 at 5-all.
She broke again to take the quarter-final clash into a deciding set.
Anisimova served for the match at 5-3 in the decider, but Andreeva kept peppering her with slices and lobs to stay alive in the duel and soon drew level.
It was Andreeva's turn to serve for the victory, but she couldn’t close at 6-5, and the contest fittingly 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."
Andreeva particularly struggled on her second serve, winning just 37 per cent of her points behind it and double-faulting seven times.
Earlier on centre court, Pegula advanced to a seventh consecutive tour-level semi-final - dating back to the 2025 US Open - with a hard-fought 6-3, 2-6, 6-4 performance against last year’s runner-up Clara Tauson.
The 31-year-old American has stepped up her game following sub-par results in Montreal and Cincinnati last summer and has since been a serious threat in the latter stages of every tournament she has played.
"My coaches and I worked on a lot of stuff to get my game back, to emphasise what I do really well, get back to the true roots of my game, just make that even better and more efficient," explained Pegula, who hit 36 winners and just 26 unforced errors against Tauson.
"I think I've been serving a lot better. Physically been feeling good. Moving better again.
"There are a lot of things. But honestly, I've been working on my game a lot. I think I've become a better player over the last six months."
In the first meeting between the pair, 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.
Pegula closed out the victory two games later to punch her ticket to the second Dubai semi-final of her career, and first since 2023.
In the third quarter-final, third-seeded Coco Gauff took down Filipina rising star Alexandra Eala 6-0, 6-2 in just over an hour to book her spot in the final four.
Gauff will next take on the winner of two-time Dubai champion Elina Svitolina and Croatian lucky loser Antonia Ruzic, who meet in the last quarter-final.