World number 102 Andreeva, who was looking to become the youngest player to reach the last eight of the grasscourt Grand Slam since Anna Kournikova in 1997, fought valiantly against Keys but was ultimately overwhelmed by the seasoned 25th seed.
The teenager's lack of experience showed towards the end of the match, as she was handed a point penalty for throwing her racket at 5-2 in the decider, which handed match point to Keys.
An emotional Andreeva then argued with the umpire, before refusing to shake hands at the end of the match.
Prior to her defeat against Keys, Andreeva had showcased great maturity in her wins over former French Open champion Barbora Krejcikova and compatriot Anastasia Potapova.
Asked if she had felt any pressure facing the breakout star, Keys said: "I'd be lying if I said I wasn't.
"Coming out here, you don't want to be the person who loses to her, for her to get to her first quarter-finals. I knew that if I tried to just stay in the match, my many, many, many more years on Tour would kick in."
Keys began the match in typically aggressive fashion, firing off powerful returns and groundstrokes to break serve and go 2-0 ahead.
However, Andreeva broke back immediately courtesy of a lucky net cord before showing great variation to upset Keys' rhythm, making her opponent uncomfortable with slices and drop shots as she broke twice more in quick succession to take the opener.
UNFORCED ERRORS
Keys continued to commit a stream of unforced errors as Andreeva raced to a 3-0 lead in the second set.
But with her back to the wall and staring at a 5-1 deficit, the American upped her game, putting the match back on serve with a delicate left-handed winner and forcing a tiebreak, which she won.
"I don't know how I turned it around. I knew she is a phenomenal player who has no pressure and is really going for everything," Keys said.
"I knew I had to stay in the match and get an opportunity to break back. Once I was able to, I tried to keep the momentum and keep going and here we are."
The tiebreak seemed to take the wind out of Andreeva's sails. Keys stormed into an early 2-0 lead in the third set after the Russian double faulted in her opening service game, before finishing off her opponent in a shade over two hours.
Keys' win ensures her return to the quarter-finals for the first time since 2015, after which she has twice been knocked out in the fourth round.
"My quarter-final run all those years ago was amazing, but I've fallen short a few times," Keys added. "It's so great to be back in the quarter-finals at Wimbledon."
In her second quarter-final at Wimbledon, Keys will face either second-seeded Belarusian Aryna Sabalenka or another Russian in Ekaterina Alexandrova, the 21st seed.
