Andreeva, who came through the qualifying tournament, defeated 22nd-seeded compatriot Anastasia Potapova (22) 6-2, 7-5.
"Even if I wanted to show my emotions, I couldn't because I was out of breath on every point," said Andreeva.
The teenager had to battle hard in the second set, coming back from 1-4 down before squandering seven break points in the ninth game.
But she held her nerve to break Potapova in the 11th game before securing victory when her opponent buried an overhead into the net.
It was Potapova's 45th and final unforced error.
"It was an amazing battle," said Andreeva. "She played really well. I was 4-1 down in the second set and I didn't feel great."
Andreeva arrived at Wimbledon having made a run to the third round at the French Open, also out of qualifying.
She will rise in the rankings from 102 to the top 65 at least after Wimbledon.
However, she has loftier ambitions and is targeting the seeded players' dressing room at Wimbledon in 2024.
"I hope to be in the locker room above next year," added the teenager, who will face US 25th seed Madison Keys on Monday for a place in the quarter-finals.
