World number 115 Tan, who stunned 23-times Grand Slam champion Serena Williams in a late-night epic on day two, backed up that win with two more impressive performances.
But her wily game was no match for 20th seed Anisimova whose powerful baseline game got the job done in 74 minutes.
Anisimova, who beat title contender Coco Gauff in the previous round and is the last remaining American woman in the singles, will take on 2019 champion Simona Halep next.
Since reaching the French Open semi-finals in 2019 and being labelled the next big thing in American women’s tennis, New Jersey-born Anisimova has experienced a traumatic time.
Her father died of a heart attack just before the U.S. Open that year and since then she has been rather overshadowed by other young players who have burst through.
The 20-year-old arrived at Wimbledon under the radar but has thrived to reach her second Grand Slam quarter-final.
She quickly got on top of Tan on Monday, crunching away a forehand to break serve in the third game.
When a weary-looking Tan looped a forehand long to trail 1-4, the writing was on the wall.
Tan, who had only ever won two Grand Slam main draw matches before this Wimbledon, stopped the rot by holding serve at 5-1.
But she never really looked capable of springing another surprise.
A double-fault gifted Anisimova an early break of serve in the second set and the American broke again to seal victory with a punishing backhand return.
“It was a really tough match today. I was a little bit tired. She played really well. But I’m really happy, you know, to be in the second week. I think it was an amazing tournament for me,”Tan said.