Boulter, 25, played out a gruelling two hours with Pliskova and overcame a set deficit to beat the runner-up of last year's tournament, entering the third round of a Grand Slam for the first time.
After the match, Boulter said: "My Gran passed away two days ago, and I'd just like to dedicate that to her today."
"I got absolutely no words right now, I'm literally shaking. The crowd was unbelievable, so thank you so much for getting me through that."
Compatriot Heather Watson will join Boulter in the third round after overcoming China's Qiang Wang, although Harriet Dart will depart this year's Championships following a defeat to the United States' Jessica Pegula.
Draper, 20, beat Belgian wildcard Zizou Bergs in the first round but fell short in a competitive second-round match which lasted over three hours.