After making a break in the opening game, Ostapenko fell behind as Cibulkova battled back to fuel her own hopes of progressing beyond the quarter-finals for the first time.
She will meet German 11th seed Angelique Kerber for a place in Saturday's final.
Up next for the 23-time Grand Slam victor is German 13th seed Julia Goerges, who also overcame an early blip in her quarter-final match to defeat close friend Kiki Bertens 3-6 7-5 6-1.
French Open champion Ostapenko, 21, was clearly the more adventurous player in her encounter with unseeded Cibulkova, pumping 33 winners and 28 unforced errors.
Twelfth seed Jelena Ostapenko recovered from 5-2 down in the first set to reach the Wimbledon quarter-finals on Monday, channelling her anger from a disputed code violation to crush Aliaksandra Sasnovich 7-6 (4) 6-0. If there is one lesson I would teach my daughter it's never give in, always fight. "I think the last game shows how good we both played and how she was fighting until the end".
"I had some great matches", she said.
Over the last week, Kerber has watched all the top 10 seeds perish, which has raised expectations that she could repeat her 2016 run to the final. "It never happened to me in my career", Cibulkova fumed. "We are here in the semis and they are all tough opponents and you have to play at your best", Kerber said.
"I did ballroom dancing from the age of five to 12", she said.
Meghan Markle to cheer Wimbledon royal Serena
Their jam-packed itinerary included meetings with the Taoiseach Leo Varadkar and Irish President Michael D. She also sported some designer accessories during the trip, including a Fendi bag worth £2,890.
Giorgi won the sixth game and closed the gap a little, but Serena had upped her intensity level and took the set with a triumphant shout. Just going to prepare well for the match.
"It was just the way I felt today".
Up next for Kerber is 2017 French Open champion Jelena Ostapenko, who reached her first Wimbledon semi-final after beating former world number four Dominika Cibulkova on Court Number One.
Kerber looked anxious and Ostapenko had a point to close to within a game but she dumped a backhand return into net and then coughed up two more errors to hand Kerber victory.
"I don't think Kasatkina has been disciplined enough". She clearly had Cibulkova beaten on winners, scoring 23 to her four, but had more than twice the number of errors to limit the difference going into the second set.
In a match that featured eight breaks of serve - four in the first five games - Ostapenko was able to elevate her game in the crucial moments.
Men's quarterfinals: Kevin Anderson beat Roger Federer 2-6, 6-7 (5), 7-5, 6-4, 13-11; John Isner beat Milos Raonic 6-7 (5), 7-6 (7), 6-4, 6-3; Novak Djokovic beat Kei Nishikori 6-3, 3-6, 6-2, 6-2; Rafael Nadal beat Juan Martin del Potro 7-5, 6-7 (7), 4-6, 6-4, 6-4.
Simon failed to convert a break point, then a third match point for Del Potro ended with him netting a forehand.