Russian seventh seed Andreeva frequently looked out of sorts while Bencic was a picture of calm in a 7-6 (7-3) 7-6 (7-2) win on Centre Court.
World number 35 and former Olympic champion Bencic served for the match at 5-4 and was two points away from the last four.
But a match that had gone blow for blow throughout was only heading one way and Bencic secured victory in a second-set tie-break.
After an emotional reaction when she booked her place in the last eight, the 28-year-old simply raised her arms in the air and smiled as she advanced again.
By contrast, 18-year-old Andreeva was in tears as she left the court having missed the chance to play in a second major semi-final.
Earlier, Poland's Swiatek had booked a maiden semi-final at the All England Club with a 6-2 7-5 victory over Russian Liudmila Samsonova.
World number one Aryna Sabalenka and American 13th seed Amanda Anisimova will contest the other semi-final.
It has been a remarkable year for Bencic, who won the Abu Dhabi Open in February just 10 months after giving birth to her daughter Bella.
But in Andreeva she faced one of the favourites for the title at the All England Club.
Having won two WTA 1,000 titles - the tier of tournaments below the majors - she has climbed to a career-high ranking of sixth in the world this year.
In a match of long, intense rallies the pair could barely be separated.
Bencic saved break points at 5-5 in the first set but was the more composed in the tie-break while Andreeva lost her way and, despite producing some rapid serves, she could not match Bencic's level.
That was the first set Andreeva had dropped at the tournament having previously cruised through her matches.
But nerves and frustration were evident from the teenager while Bencic quietly went under the radar with an assured performance.
Andreeva saved break points at 2-1 in the second set and Bencic did the same at 4-4 before finding the crucial break of serve to give herself an opportunity to serve out the match.
Andreeva immediately responded to force what felt like an inevitable tie-break as her coach Conchita Martinez – a former Wimbledon champion herself – watched on with an anxious look on her face in the coaching box.
But Andreeva's nerves seemed to get the better of her and a double fault handed Bencic four match points and she took her chance at the second opportunity.
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