The Pole, who is playing her only grasscourt tournament in the run-up to Wimbledon, which starts next week, struggled at times and had 37 unforced errors against Alexandrova before setting up a semi-final against second-seed Jasmine Paolini.
"I think today was all about adjusting to the conditions because it was super windy," Swiatek said.
"On one hand, to be intense, on the other hand, being safe."
"Ekaterina is a tough opponent. I have lost against her before. She can play amazing tennis, and her flat shots really fit the grass, so I am happy I am through."
She won the first set after breaking Alexandrova twice, but the match was interrupted midway through the second set with rain starting to fall.
Swiatek said she kept focused by solving crossword puzzles during the break, and it seemed to work as she held serve throughout the second set to clinch the tiebreak and face Paolini next. Swiatek currently leads Paolini in their head-to-head 4-0.
The Italian, a Wimbledon finalist last year, recorded her fourth win over Beatriz Haddad Maia in as many meetings with a 7-5, 7-5 victory.
It was also her first win over the Brazilian in straight sets as the Italian reached her fourth semi-final of the season.
Top-seed Jessica Pegula joined her in the last four after needing three sets to edge past fellow American and fifth-seed Emma Navarro 6-4, 1-6, 6-3.
Pegula will face 20-year-old Czech Linda Noskova, who outplayed third-seed Mirra Andreeva 6-3, 6-3.