Vondrousova looked uncomfortable throughout the contest, committing 44 unforced errors, but held her nerve to grind out a win despite dropping the first set.
"It was a very tough match. We played a long match, and we know each other from the very young age, so it was very tough also mentally," Vondrousova told reporters.
"I'm just very happy that I stayed in the match and I stayed focused. I just wanted to stay in the match and be more active and going for the volleys and everything."
Bouzkova started well, winning four straight games en route to a 5-1 lead in the opening set as several of Vondrousova's errant shots found the net.
The 32nd seed stumbled while serving for the set, allowing Vondrousova to reclaim a break, but she broke back immediately to seal the opener.
Vondrousova launched her comeback in the second set as she produced some deft drop shots and bravely approached the net to claim two breaks and force a decider.
With her confidence restored, she made no mistake as she moved 4-3 ahead and then wrapped up victory with another break.
"I didn't get many wins on grass before. Now I'm in a quarter-finals, so I wasn't expecting that," said Vondrousova, who had failed to progress past Wimbledon's second round in four previous attempts as a singles player.
"I talked with my coaches before and they told me 'You have the game for it'. I think you just have to believe it, you know, be open-minded and just work for it."
Vondrousova will meet the winner of the tie between fourth seed Jessica Pegula and Ukraine's Lesia Tsurenko for a place in the semi-finals.